22 Jan 2021 Length Functions on Groups and Rigidity

Home , Quotient group, Superrigidity

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• (The stable word lengths) Let G be a group generated by a symmetric (not necessar-

ily finite) set S. For any g ∈ G, the word length φS(w) = min{n | g = s1s2 ··· sn,each si ∈ S} is the minimal number of elements of S whose product is g. The stable length is defined as φ (gn) l(g) = lim S . n→∞ n • (Stable norms) Let M be a compact smooth manifold and G = Diff(M) the diffeomorphism group consisting of all self-diffeomorphisms. For any diffeomorphism f : M → M, let kfk = sup kDxfk, x∈M

where Dxf is the induced linear map between tangent spaces TxM → Tf(x)M. Deﬁne

log kf nk log kf −nk l(f) = max{ lim , lim }. n→+∞ n n→+∞ n

• (smooth measure-theoretic entropy) Let M be a C∞ closed Riemannian manifold 2 and G = Diffµ(M) consisting of diffeomorphisms of M preserving a Borel probability measure µ. Let l(f) = hµ(f) be the measure-theoretic entropy, for any f ∈ G = 2 Diffµ(M). • (Translation lengths) Let (X,d) be a metric space and G = Isom(X) consisting of isometries γ : X → X. Fix x ∈ X, define d(x, γnx) l(γ) = lim . n→∞ n This contains the translation lengths on CAT(0) spaces and Gromov δ-hyperbolic spaces as special cases.

• (average norm for quasi-cocycles) Let G be a group and E be a Hilbert space with an G-action by linear isometrical action. A function f : G → E is a quasi-cocyle if there exists C > 0 such that

kf(gh) − f(g) − gf(h)k

for any g, h ∈ G. Let l : G → [0, +∞) be deﬁned by

kf(gn)k l(g) = lim . n→∞ n

• (Rotation numbers of circle homeomorphisms) Let R be the real line and G = HomeZ(R) = {f | f : R → R is a monotonically increasing homeomorphism such that f(x + n) = f(x) for any n ∈ Z}. For any f ∈ HomeZ(R) and x ∈ [0, 1), the translation number is deﬁned as f n(x) − x l(f) = lim . n→∞ n

2 • (Asymptotic distortions) Let f be a C1+bv diffeomorphism of the closed interval [0, 1] or the circle S1. (“bv” means derivative with finite total variation.) The asymptotic distortion of f is defined (by Navas [50]) as l(f) = lim var(log Df n). n→∞ This gives a length function l on the group Diff1+bv(M) of C1+bv diffeomorphisms for M = [0, 1] or S1. • (Dynamical degree) Let CP n be the complex projective space and f : CP n 99K CP n be a birational map given by

(x0 : x1 : ··· : xn) 99K (f0 : f1 : ··· : fn),

where the fi’s are homogeneous polynomials of the same degree without common factors. The degree of f is deg f = deg fi. Define 1 1 l(f) = max{ lim log deg(f n) n , lim log deg(f −n) n }. n→∞ n→∞ This gives a length function l : Bir(CP n) → [0, +∞). Here Bir(CP n) is the group of birational maps, also called Cremona group. The terminologies of length functions are used a lot in the literature (eg. [28], [22]). However, they usually mean different things from ours (in particular, it seems that the condition 3) has not been addressed for commuting elements before). Our first observation is the following result on vanishing of length functions.

2 Theorem 0.1 Let GA = Z ⋊A Z be an abelian-by-cyclic group, where A ∈ SL2(Z).

2 (i) When the absolute value of the trace |tr(A)| > 2, any length function l : Z ⋊A Z → 2 R≥0 vanishes on Z .

2 (ii) When |tr(A)| = 2 and A =6 I2, any length function l : Z ⋊A Z → R≥0 vanishes on the direct summand of Z2 spanned by eigenvectors of A.

2 Corollary 0.2 Suppose that the semi-direct product GA = Z ⋊A Z acts on a compact manifold by Lipschitz homeomorphisms (or C2-diﬀeomorphisms, resp.). The topological 2 entropy htop(g)=0 (or Lyapunov exponents of g are zero, resp.) for any g ∈ Z when |tr(A)| > 2 or any eigenvector g ∈ Z2 when |tr(A)| =2.

It is well-known that the central element in the integral Heisenberg group GA (for 1 1 A = ) is distorted in the word metric. When the Heisenberg group G acts on a 0 1 A C∞ compact Riemannian manifold, Hu-Shi-Wang [36] proves that the topological entropy and all Lyapunov exponents of the central element are zero. These results are special cases of Theorem 0.1 and Corollary 0.2, by choosing special length functions. A length function l : G → [0, ∞) is called purely positive if l(g) > 0 for any inﬁnite- order element g. A group G is called virtually poly-positive, if there is a ﬁnite-index subgroup H

1= Hn ⊳ Hn−1 ⊳ ··· ⊳ H0 = H such that every ﬁnitely generated subgroup of each quotient Hi/Hi+1 (i = 0, ..., n − 1) has a purely positive length function. Our following results are on the rigidity of group homomorphisms.

3 Theorem 0.3 Let Γ be an arithmetic subgroup of a simple algebraic Q-group of Q-rank at least 2. Suppose that G is virtually poly-positive. Then any group homomorphism f :Γ → G has its image ﬁnite.

Theorem 0.4 Let G be a group having a ﬁnite-index subgroup H

(i) every ﬁnitely generated subgroup of each quotient Hi/Hi+1 (i = 0, ..., n − 1) has a purely positive length function, i.e. G is virtually poly-positive; and

(ii) any torsion abelian subgroup in every ﬁnitely generated subgroup of each quotient Hi/Hi+1 (i =0, ..., n − 1) is ﬁnitely generated.

Let R be a finitely generated associative ring with identity and En(R) the elementary subgroup. Suppose that Γ < En(R) is finite-index subgroup. Then any group homomorphism f :Γ → G has its image finite when n ≥ 3.

Corollary 0.5 Let Γ be an arithmetic subgroup of a simple algebraic Q-group of Q-rank at least 2, or a ﬁnite-index subgroup of the elementary subgroup En(R) (n ≥ 3) for an associative ring R. Then any group homomorphism f : E → G has its image ﬁnite. Here G is one of the following groups:

• a Gromov hyperbolic group,

• CAT(0) group,

• automorphism group Aut(Fk) of a free group,

• outer automorphism group Out(Fk) of a free group or

• mapping class group MCG(Σg) (g ≥ 2). • the group Diﬀ(Σ,ω) of diﬀeomorphisms of a closed surface preserving an area form ω.

Theorem 0.6 Suppose that G is virtually poly-positive. Let R be a finitely generated associative ring of characteristic zero such that any nonzero ideal is of a finite index (eg. the ring of algebraic integers in a number field). Suppose that S < En(R) is a finite-index subgroup of the elementary group. Then any group homomorphism f : S → G has its image finite when n ≥ 3.

Corollary 0.7 Let R be an associative ring of characteristic zero such that any nonzero ideal is of a finite index. Any group homomorphism f : E → G has its image finite, where E < En(R) is finite-index subgroup and n ≥ 3. Here G is one of the followings:

• Aut(Fk), Out(Fk), MCG(Σg), • a hyperbolic group,

• a CAT(0) group or more generally a semi-hyperbolic group,

4 • a group acting properly semi-simply on a CAT(0) space, or

• a group acting properly semi-simply on a δ-hyperbolic space,

• the group Diﬀ(Σ,ω) of diﬀeomorphisms of a hyperbolic closed surface preserving an area form ω.

Some relevent cases of Theorem 0.4 and Theorem 0.6 are already established in the literature. Bridson and Wade [17] showed that any group homomorphism from an irreducible lattice in a semisimple Lie group of real rank ≥ 2 to the mapping class group MCG(Σg) must have its image finite. However, Theorem 0.3 can never hold when Γ is a cocompact lattice, since a cocompact lattice has its stable word length purely positive. When the length functions involved in the virtually poly-positive group G are required to be stable word lengths, Theorem 0.3 holds more generally for Γ non-uniform irreducible lattices a semisimple Lie group of real rank ≥ 2 (see Proposition 8.4). When the length functions involved in the virtually poly-positive group G are given by a particular kind of quasi-cocyles, Theorem 0.3 holds more generally for Γ with property TT (cf. Py [54], Prop. 2.2). Haettel [33] prove that any action of a high-rank a higher rank lattice on a Gromov-hyperbolic space is elementary (i.e. either elliptic or parabolic). Guirardel and Horbez [33] prove that every group homomorphism from a high-rank lattice to the outer automorphism group of torsion-free hyperbolic group has finite image. Thom [57] (Corol- lary 4.5) proves that any group homomorphism from a boundedly generated with property (T) to a Gromov hyperbolic group has finite image. Compared with these results, our target group G and the source group En(R) (can be defined over any non-commutative ring) in Theorem 0.4 are much more general. The inequalities of n in Theorem 0.4, Theo- rem 0.6 and Corollary 0.5, Corollary 0.7 can not be improved, since SL2(Z) is hyperbolic. The group Γ in Corrolary 0.5 has Kazhdan’s property T (i.e. an arithmetic subgroup of a simple algebraic Q-group of Q-rank at least 2, or a finite-index subgroup of the elementary subgroup En(R), n ≥ 3, for an associative ring R has Kazhdan’s property T by [23]). However, there exist hyperbolic groups with Kazhdan’s property T (cf. [32], Section 5.6). This implies that Corrolary 0.5 does not hold generally for groups Γ with Kazhdan’s property T. Franks and Handel [27] prove that any group homomorphism from a quasi-simple group containing a subgroup isomorphic to the three-dimensional integer Heisenberg group, to the group Diff(Σ,ω) of diffeomorphisms of a closed surface preserving an area form ω, has its image finite (cf. Lemma 8.3).

We now study length functions on the Cremona groups.

n Theorem 0.8 Let Bir(Pk )(n ≥ 2) be the group of birational maps on the projective space n n Pk over an algebraic closed ﬁeld k. Any length function l : Bir(Pk ) → [0, +∞) vanishes n on the automorphism group Aut(Pk ) = PGLn+1(k).

When n =2, a result of Blanc and Furter [9] (page 7 and Proposition 4.8.10) implies 2 n that there are three length functions l1,l2,l3 on Bir(Pk) such that any element g ∈ Bir(Pk ) 2 satisfying l1(g)= l2(g)= l3(g) = 0 is either ﬁnite or conjugate to an element in Aut(Pk). n This implies that the automorphism group Aut(Pk ) (when k = 2) is one of the ‘largest’ n subgroups of Bir(Pk ) on which every length function vanishes.

Corollary 0.9 Let G be a virtually poly-positive group. Any group homomorphism f : 2 Bir(Pk) → G is trivial, for an algebraic closed ﬁeld k.

5 2 In particular, Corrolary 0.9 implies that any quotient group of Bir(Pk) cannot act properly semisimply neither on a Gromov δ-hyperbolic space nor a CAT(0) space. This is interesting, considering the following facts. There are (inﬁnite-dimensional) hyperbolic 2 space and cubical complexes, on which Bir(Pk ) acts isometrically (see [21], Section 3.1.2 2 and [45]). The Cremona group Bir(Pk) is sub-quotient universal: every countable group 2 can be embedded in a quotient group of Bir(Pk) (see [21], Theorem 4.7). Moreover, Blanc- Lamy-Zimmermann [10] (Theorem E) proves that when n ≥ 3, there is a surjection from n Bir(Pk ) onto a free product of two-element groups Z/2. This means that Corrolary 0.9 can never hold for higher dimensional Cremona groups.

As byproducts, we give characterizations of length functions on Lie groups. Our next result is that there is essentially only one length function on the special linear group SL2(R):

Theorem 0.10 Let G = SL2(R). Any length function l : G → [0, +∞) continuous on the subgroup SO(2) and the diagonal subgroup is proposional to the translation function

τ(g) := inf d(x, gx), x∈X where X = SL2(R)/SO(2) is the upper-half plane.

More generally, we study length functions on Lie groups. Let G be a connected semisimple Lie group whose center is ﬁnite with an Iwasawa decomposition G = KAN. Let W be the Weyl group, i.e. the quotient group of the normalizers NK (A) modulo the centralizers CK (A). Our second result shows that a length function l on G is uniquely determined by its image on A.

Theorem 0.11 Let G be a connected semisimple Lie group whose center is ﬁnite with an Iwasawa decomposition G = KAN. Let W be the Weyl group.

(i) Any length function l on G that is continuous on the maximal compact subgroup K is determined by its image on A.

(ii) Conversely, any length function l on A that is W -invariant (i.e. l(w · a) = l(a)) can be extended to be a length function on G that vanishes on the maximal compact subgroup K.

The proofs of Theorems 0.10 0.11 are based the Jordan-Chevalley decompositions of algebraic groups and Lie groups. We will prove that any length function on a Heisenberg group vanishes on the central elements (see Lemma 5.2). This is a key step for many other proofs. Based this fact, we prove Theorems 0.3, 0.6, 0.4, 0.8 by looking for Heisenberg subgroups. In Section 1, we give some elementary facts on the length functions. In Section 2, we discuss typical examples of length functions. In later sections, we study length functions on Lie groups, algebraic groups, hyperbolic groups, matrix groups and the Cremona groups.

6 1 Basic properties of length functions

1.1 Length functions Deﬁnition 1.1 Let G be a group. A function l : G → [0, ∞) is called a length function if 1) l(gn)= |n|l(g) for any g ∈ G and n ∈ Z. 2) l(aga−1)= l(g) for any a, g ∈ G. 3) l(ab) ≤ l(a)+ l(b) for commuting elements a, b.

Lemma 1.2 Any torsion element g ∈ G has length l(g)=0.

Proof. Note that l(1) = 2l(1) and thus l(1) = 0. If gn =1, then l(g)= |n|l(1) = 0. Recall that a subset V of a real vector space is a convex cone, if av + bw ∈ V for any v,w ∈ V and any non-negative real numbers a, b ≥ 0.

Lemma 1.3 The set Func(G) of all length functions on a group G is a convex cone.

Proof. It is obvious that for two functions l1,l2 on G, a non-negative linear combination al1 + bl2 is a new length function. Lemma 1.4 Let f : G → H be a group homomorphism between two groups G and H. For any length function l : H → [0, ∞), the composite l ◦ f is a length function on G.

Proof. It is enough to note that a group homomorphism preserves powers of elements, conjugacy classes and commutativity of elements.

Corollary 1.5 For a group G, let Out(G) = Aut(G)/Inn(G) be the outer automorphism group. Then Out(G) acts on the set Func(G) of all length functions by pre-compositions l 7→ l ◦ g, where l ∈ Func(G), g ∈ Out(G). This action preserves scalar multiplications and linear combinations (with non-negative coeﬃcients).

−1 Proof. For an inner automorphism Ig : G → G given by Ig(h)= ghg , the length function l ◦ Ig = l since l is invariant under conjugation. Therefore, the outer automorphism group Out(G) has an action on Func(G). It is obvious that the pre-compositions preserve scalar multiplications and linear combinations with non-negative coeﬃcients.

Deﬁnition 1.6 A length function l : G → [0, ∞) is primitive if it is not a composite l′ ◦f for a non-trivial surjective group homomorphism f : G ։ H and a length function l′ : H → [0, ∞).

Lemma 1.7 Suppose that a length function l : G → [0, ∞) vanishes on a central subgroup H < G. Then l factors through the quotient group G/H. In other words, there exists a length function l′ : G/H → [0, ∞) such that l = l′ ◦ q, where q : G → G/H is the quotient group homomorphism.

Proof. Write G = ∪gH, the union of left cosets. For any h ∈ H, we have l(gh) ≤ l(g)+ l(h) = l(g) and l(g) = l(ghh−1) ≤ l(gh). Therefore, l(gh) = l(g) for any h ∈ H. Deﬁne l′(gH)= l(g). Then l′ is a length function on the quotient group G/H. The required property follows the deﬁnition easily.

7 Corollary 1.8 Suppose that a group G has non-trivial ﬁnite central subgroup Z(G). Any length function l on G factors through G/Z(G).

Proof. This follows Lemma 1.7 and Lemma 1.2.

Lemma 1.9 Let G be a group. Suppose that any non-trivial normal subgroup H ⊳ G is of ﬁnite index. Then any non-vanishing length function l : G → [0, ∞) is primitive.

Proof. Suppose that l is a composite l′ ◦ f for a non-trivial surjective group homomorphism f : G ։ H and a length function l′ : H → [0, ∞). By the assumption of G, the quotient group H is ﬁnite. This implies that l′ and thus l vanishes, which is a contradiction.

Theorem 1.10 Let Γ be an irreducible lattice in a connected irreducible semisimple Lie group of real rank ≥ 2. Then any non-vanishing length function l : Γ → [0, ∞) factors through a primitive function on Γ/Z(Γ).

Proof. By the Margulis-Kazhdan theorem (see [62], Theorem 8.1.2), any normal subgroup N of Γ either lies in the center of Γ (and hence it is ﬁnite) or the quotient group Γ/N is ﬁnite. Corollary 1.8 implies that l factors through a length function l′ on Γ/Z(Γ). The previous lemma 1.9 implies that l′ is primitive.

2 Examples of length functions

Let’s see a general example ﬁrst. Let G be a goup and f : G → [0, +∞) be a function satisfying f(gh) ≤ f(g)+ f(h) and f(g) = f(g−1) for any elements g, h ∈ G. Deﬁne l : G → [0, +∞) by f(gn) l(g) = lim n→∞ n for any g ∈ G.

Lemma 2.1 The function l is a length function in the sense of Deﬁnition 1.1.

Proof. For any g ∈ G,and natural numbers n, m, we have f(gn+m) ≤ f(gn)+f(gm). This n n ∞ f(g ) means that {f(g )}n=1 is a subadditive sequence and thus the limit limn→∞ n exists. This shows that l is well-deﬁned. From the deﬁnition of l, it is clear that l(gn) = |n|l(g) for any integer n. Let h ∈ G. We have f(hgnh−1) f(h)+ f(gn)+ f(h−1) f(gn) l(hgh−1) = lim ≤ lim = lim = l(g). n→∞ n n→∞ n n→∞ n Similarly, we have l(g) = l(h−1(hgh−1)h) ≤ l(hgh−1) and thus l(g) = l(hgh−1). For commuting elements a, b, we have (ab)n = anbn. Therefore, f((ab)n) f(anbn) l(ab) = lim = lim n→∞ n n→∞ n f(an)+ f(bn) ≤ lim ≤ l(a)+ l(b). n→∞ n

Many (but not all) length functions l come from subadditive functions f.

8 2.1 Stable word lengths Let G be a group generated by a (not necessarily ﬁnite) set S satisfying s−1 ∈ S for each s ∈ S. For any g ∈ G, the word length φS(w) = min{n | g = s1s2 ··· sn,each si ∈ S} is the minimal number of elements of S whose product is g. The stable length n φS (g ) n l(g) = limn→∞ n . Since φS(g ) is subadditive, the limit always exists. Lemma 2.2 The stable length l : G → [0, +∞) is a length function in the sense of Deﬁnition 1.1.

Proof. From the deﬁnition of the word length φS, it is clear that φS(gh) ≤ φS(g)+ φS(h) −1 and φS(g)= φS(g ) for any g, h ∈ G. The claim is proved by Lemma 2.1. When S is the set of commutators, the l(g) is called the stable commutator length, which is related to lots of topics in low-dimensional topology (see Calegari [20]).

2.2 Growth rate Let G be a group generated by a finite set S satisfying s−1 ∈ S for each s ∈ S. Sup- pose |·|S is the word length of (G,S). For any automorphism α : G → G, define ′ n ′ log l (α ) l (α) = max{|α(si)|S : si ∈ S}. Let l(α) = limn→∞ n . This number l(α) is called the algebraic entropy of α (cf. [40], Definition 3.1.9, page 114). Lemma 2.3 Let Aut(G) be the group of automorphisms of G. The function l : Aut(G) → [0, +∞) is a length function in the sense of Definition 1.1.

−1 −1 ′ ′ −1 Proof. Since α(si) = α (si) for any si ∈ S, we know that l (α) = l (α ). For ′ another automorphism β : G → G, let l (β) = |β(si)|S for some si ∈ S. Suppose that ′ ′ β(si)= si1 si2 ··· sik with k = l (β). Then |(αβ)(si)|S = |α(si1 )α(si2 ) ··· α(sik )|S ≤ l (α)k. This proves that l′(αβ) ≤ l′(α)l′(β). The claim is proved by Lemma 2.1.

n Fix g ∈ G. For any automorphism α : G → G, deﬁne bn = |α (g)|S. Suppose that n n n n g = s1s2 ··· sk with k = |g|S. Note that bn = |α (g)|S = |α (s1)α (s2) ··· α (sk)|S ≤ ′ n l (α )|g|S. Therefore, we have log b lim sup n ≤ l(α). n→∞ n n This implies that l(α) is an upper bounded for growth rate of {|α (g)|S}. The growth rate is studied a lot in geometric group theory (for example, see [43] for growth of automorphisms of free groups).

2.3 Matrix norms and group acting on smooth manifolds

For a square matrix A, the matrix norm kAk = supkxk=1 kAxk. Deﬁne the stable norm log kAnk s(A) = limn→+∞ n . Since kABk≤kAkkBk for any two matrices A, B, the sequence n ∞ {log kA k}n=1 is subadditive and thus the limit exists.

Lemma 2.4 Let G = GLn(R) be the general linear group. The function l : G → [0, +∞) deﬁned by l(g) = max{s(g),s(g−1)} is a length function in the sense of Deﬁnition 1.1.

9 Proof. From the definition of the matrix norm, it is clear that log kghk ≤ log kgk+log khk for any g, h ∈ G. Then l(g) = max{s(g),s(g−1)} is a length function by Lemma 2.1. Let M be a compact smooth manifold and Diff(M) the diffeomorphism group consisting of all self-diffeomorphisms. For any diffeomorphism f : M → M, let

kfk = sup kDxfk, x∈M where Dxf is the induced linear map between tangent spaces TxM → Tf(x)M. Deﬁne

log kf nk log kf −nk l(f) = max{ lim , lim }. n→+∞ n n→+∞ n A similar argument as the proof of the previous lemma proves the following.

Lemma 2.5 Let G be a group acting on a smooth manifold M by diﬀeomorphisms. The function l : G → [0, +∞) is a length function in the sense of Deﬁnition 1.1.

For an f-invariant Borel probability measure µ on M, it is well known (see [O]) that there exists a measurable subset Γf ⊂ M with µ(Γf ) = 1 such that for all x ∈ Γf and u ∈ TxM, the limit 1 χ(x, u, f) = lim log kD f n(u)k n x exists and is called Lyapunov exponent of u at x. From the deﬁnitions, we know that χ(x, u, f) ≤ l(f) for any x ∈ Γf and u ∈ TxM.

2.4 Smooth measure-theoretic entropy Let T : X → X be a measure-preserving map of the probability space (X, B, m). For a ﬁnite-sub-σ-algebra A = {A1, A2, ..., Ak} of B, denote by

H(A) = − m(Ai) log m(Ai), X1 h(T, A) = lim H(∨n−1T −iA), n i=0

n−1 −i n−1 −i where ∨i=0 T A is a set consisting of sets of the form ∩i=0 T Aji . The entropy of T is deﬁned as hm(T ) = sup h(T, A), where the supremum is taken over all ﬁnite sub-algebra A of B. For more details, see Walters [58] (Section 4.4).

∞ 2 Lemma 2.6 Let M be a C closed Riemannian manifold and G = Diffµ(M) consisting of diffeomorphisms of M preserving a Borel probability measure µ. The entropy hµ is a 2 length function on Diffµ(M) in the sense of Definition 1.1.

2 n Proof. For any f,g ∈ Diﬀµ(M) and integer n, it is well-known that hµ(f ) = |n|hµ(f) −1 and hµ(f) = hµ(gfg ) (cf. [58], Theorem 4.11 and Theorem 4.13). Hu [35] proves that hµ(fg) ≤ hµ(f)+ hµ(g) when fg = gf.

10 2.5 Stable translation length on metric spaces Let (X,d) be a metric space and γ : X → X an isometry. Fix x ∈ X. Note that −1 d(x, γ1γ2x) ≤ d(x, γ1x)+d(γ1x, γ1γ2x)= d(x, γ1x)+d(x, γ2x) and d(x, γ1x)= d(x, γ1 x) for any isometries γ1,γ2. Deﬁne d(x, γnx) l(γ) = lim . n→∞ n For any y ∈ X, we have

d(x, γnx) ≤ d(x, y)+ d(y,γny)+ d(γny,γnx) = 2d(x, y)+ d(y,γny)

d(x,γnx) d(y,γny) and thus limn→∞ n ≤ limn→∞ n . Similarly, we have the other direction d(y,γny) d(x, γnx) lim ≤ lim . n→∞ n n→∞ n This shows that the deﬁnition of l(γ) does not depend on the choice of x.

Lemma 2.7 Let G be a group acting isometrically on a metric space X. Then the function l : G → [0, +∞) deﬁned by g 7−→ l(g) as above is a length function in the sense of Deﬁnition 1.1.

Proof. This follows Lemma 2.1.

2.6 Translation lengths of isometries of CAT(0) spaces In this subsection, we will prove that the translation length on a CAT(0) space deﬁnes a length function. First, let us introduce some notations. Let (X,dX ) be a geodesic metric space, i.e. any two points x, y ∈ X can be connected by a path [x, y] of length dX (x, y). For three points x, y, z ∈ X, the geodesic triangle ∆(x, y, z) consists of the three vertices x, y, z and the three geodesics [x, y], [y, z] and [z, x]. Let R2 be the Euclidean plane with 2 the standard distance dR2 and ∆¯ a triangle in R with the same edge lengths as ∆. Denote by ϕ : ∆ → ∆¯ the map sending each edge of ∆ to the corresponding edge of ∆¯ . The space X is called a CAT(0) space if for any triangle ∆ and two elements a, b ∈ ∆, we have the inequality dX (a, b) ≤ dR2 (ϕ(a),ϕ(b)). The typical examples of CAT(0) spaces include simplicial trees, hyperbolic spaces, prod- ucts of CAT(0) spaces and so on. From now on, we assume that X is a complete CAT(0) space. Denote by Isom(X) the isometry group of X. For any g ∈ Isom(X), let

Minset(g)= {x ∈ X : d(x, gx) ≤ d(y,gy) for any y ∈ X} and let τ(g) = infx∈X d(x, gx) be the translation length of g. When the ﬁxed-point set Fix(g) =6 ∅, we call g elliptic. When Minset(g) =6 ∅ and dX (x, gx) = τ(g) > 0 for any x ∈ Minset(g), we call g hyperbolic. The group element g is called semisimple if the minimal set Minset(g) is not empty, i.e. it is either elliptic or hyperbolic. A subset C of a CAT(0) space if convex, if any two points x, y ∈ C can connected by the geodesic segment [x, y] ⊂ C. A group G is called CAT(0) if G acts properly discontinuously and

11 cocompactly on a CAT(0) space X. In such a case, any inﬁnite-order element in G acts hyperbolically on X. For more details on CAT(0) spaces, see the book of Bridson and Haeﬂiger [16]. The following was proved by Ballmann-Gromov-Schroeder [2] (Lemma 6.6, page 83). The original proof was for Hardmard manifolds, which also holds for general cases. For completeness, we give details here.

Lemma 2.8 Let γ : X → X be an isometry of a complete CAT(0) space X. For any x0 ∈ X, we have d(γkx , x ) τ(γ) := inf d(γx, x) = lim 0 0 . x∈X k→∞ k

Proof. For any p = x0 ∈ X, let m be the middle point of [p,γp]. We have that d(m, γm) ≤ 1 2 2 2 2 d(p,γ p) by the convexity of length functions. Therefore, d(p,γ p) ≥ 2τ(γ) and τ(γ ) ≥ 2τ(γ). Note that d(p,γ2p) ≤ d(p,γp)+ d(γp,γ2p)=2d(p,γp) and thus τ(γ2) ≤ 2τ(γ). Inductively, we have n 2nτ(γ) ≤ d(p,γ2 p) ≤ 2nd(p,γp).

d(γkp,p) Note that the limit limk→∞ k exists and is independent of p (see the previous sub- d(γkp,p) section). Therefore, the limit limk→∞ k equals to τ(γ).

Corollary 2.9 Let X be a complete CAT(0) space and G a group acting on X by isometries. For any g ∈ G, deﬁne τ(g) = infx∈X d(x, gx) as the translation length. Then τ : G → [0, +∞) is a length function in the sense of Deﬁnition 1.1.

Proof. This follows Lemma 2.8 and Lemma 2.7.

2.7 Translation lengths of Gromov δ-hyperbolic spaces Let δ > 0. A geodesic metric space X is called Gromov δ-hyperbolic if for any geodesic triangle ∆xyz one side [x, y] is contained a δ-neighborhood of the other two edges [x, z] ∪ n [y, z]. Fix x0 ∈ X. Any isometry γ : X → X is called elliptic if {γ x}neZ is bounded. If n the orbit map Z → X given by n 7→ γ x0 is quasi-isometric (i.e. there exists A ≥ 1 and B ≥ 0 such that 1 |n − m| − B ≤ d (γnx ,γmx ) ≤ A|n − m| + B A X 0 0 for any integers n, m), we call γ is hyperbolic. Otherwise, we call γ is parabolic. Define n d(γ x0,x0) l(γ) = limn→∞ n . For any group G acts isometrically on a δ-hyperbolic space, the function l : G → [0, ∞) is a length function by Lemma 2.7. A finitely generated group G is Gromov δ-hyperbolic if for some finite generating set S, the Caley graph Γ(G,S) is Gromov δ-hyperbolic. Any infinite-order element g in a Gromov δ-hyperbolic group is hyperbolic and thus has positive length l(g) > 0 (cf. [32], 8.1.D). For more details on hyperbolic spaces and hyperbolic groups, see the book [32] of Gromov.

12 2.8 Quasi-cocycles Let G be a group and (E, k·k) be a normed vector space with an G-action by linear isometries. A function f : G → E is a quasi-cocyle if there exists C > 0 such that

kf(gh) − f(g) − gf(h)k

Lemma 2.10 For any quasi-cocyle f : G → E, the average norm l is a length function.

Proof. For any natural number n, we have

kf(1) − f(g−n) − g−nf(gn)k

−n −n n −n n f(1)−f(g )−g f(g ) C kf(g )k kf(g )k and thus k n k < n . Taking the limit, we have limn→∞ n = limn→∞ n . kn k kf(g )k Therefore, for any k ∈ Z, we have l(g ) = limn→∞ n = |k|l(g). For any h ∈ G, we have kf(hgnh−1)k≤kf(h)k + kf(h−1)k + kf(gn)k +2C. Therefore, we have kf(hgnh−1)k kf(gn)k l(hgh−1) = lim ≤ lim = l(g). n→∞ n n→∞ n Similarly, we have l(g)= l(h−1(hgh−1)h) ≤ l(hgh−1). When g, h commutes, we have kf((gh)n)k kf(gnhn)k l(gh) = lim = lim n→∞ n n→∞ n kf(gn)k + kf(hn)k + C ≤ lim = l(g)+ l(h). n→∞ n

2.9 Rotation number

Let R be the real line and HomeZ(R) = {f | f : R → R is a monotonically increasing homeomorphism such that f(x + n)= f(x) for any n ∈ Z}. For any f ∈ HomeZ(R) and x ∈ [0, 1), the translation number is deﬁned as f n(x) − x l(f) = lim . n→∞ n It is well-known that l(f) exists and is independent of x (see [51], Prop. 2.22, p.31). Note that every f ∈ HomeZ(R) induces an orientation-preserving homeomorphism of the circle S1.

13 Proposition 2.11 The absolute value of the translation number |l| : HomeZ(R) → [0, ∞) is a length function in the sense of Deﬁnition 1.1.

Proof. For any f ∈ HomeZ(R) and k ∈ Z\{0}, we have that f kn(x) − x f kn(x) − x l(f k) = lim = k lim = kl(f). n→∞ n n→∞ nk

For any a ∈ HomeZ(R), we have that af n(a−1x) − x − f n(x)+ x | l(afa−1) − l(f) |= lim | | n→∞ n af n(a−1x) − f n(a−1x)+ f n(a−1x) − f n(x) = lim | | n→∞ n = 0, since a is bounded on [0, 1] and | f n(a−1x) − f n(x) |≤ 2+ |a−1x − x|. For commuting elements f,g ∈ HomeZ(R), we have that f n(gn(x)) − x f n(gn(x)) − gn(x)+ gn(x) − x l(fg) = lim = lim . n→∞ n n→∞ n n Suppose that g (x)= kn + xn for kn ∈ Z and xn ∈ [0, 1). Then f n(gnx) − gn(x)+ gn(x) − x lim n→∞ n f n(0) − 0+ gn(x) − x = lim = l(f)+ l(g). n→∞ n Therefore, we get |l(fg)|≤|l(f)| + |l(g)|.

Remark 2.12 It is actually true that the rotation number l is multiplicative on any amenable group (see [51], Prop. 2.2.11 and the proof of Prop. 2.2.10, page 36). This implies that the absolute rotation number |l| is subadditive on any amenable group. In other words, for any amenable group G< HomeZ(R) and any g, h ∈ G we have |l(gh)| ≤ |l(g)| + |l(h)|.

2.10 Asymptotic distortions Let f be a C1+bv diffeomorphism of the closed interval [0, 1] or the circle S1. (“bv” means derivative with finite total variation.) The asymptotic distortion of f is defined as

n l(f) = dist∞(f) = lim var(log Df ). n→∞ It’s proved by Eynard-Bontemps amd Navas ([24], pages 7-8) that

n (1) dist∞(f )= |n|dist∞(f) for all n ∈ Z;

−1 1+bv (2) dist∞(hfh ) = dist∞(f) for every C diﬀeomorphism h;

(3) dist∞(f ◦ g) ≤ dist∞(f) + dist∞(g) for commuting f, g. Therefore, the asymptotic distortion is a length function l on the group Diﬀ1+bv(M) of C1+bv diﬀeomorphisms for M = [0, 1] or S1.

14 2.11 Dynamical degrees of Cremona groups

n n+1 Let k be a ﬁeld and Pk = k \{0}/{λ ∼ λx : λ =06 } be the projective space. A rational n map from Pk to itself is a map of the following type

(x0 : x1 : ··· : xn) 99K (f0 : f1 : ··· : fn) where the fi’s are homogeneous polynomials of the same degree without common fac- n tor. The degree of f is deg f = deg fi. A birational map from Pk to itself is a ra- n 99K n n 99K n tional map f : Pk Pk such that there exists a rational map g : Pk Pk such n that f ◦ g = g ◦ f = id. The group Bir(Pk ) of birational maps is called the Cremona n group (also denoted as Crn(k)). It is well-known that Bir(Pk ) is isomorphic to the group Autk(k(x1, x2, ··· , xn)) of self-isomorphisms of the field k(x1, x2, ··· , xn) of the rational n functions in n indeterminates over k. The (first) dynamical degree λ(f) of f ∈ Bir(Pk ) is defined as 1 1 λ(f) = max{ lim deg(f n) n , lim deg(f −n) n }. n→∞ n→∞ 1 Since deg(f n) n is sub-multiplicative, the limit exists. n Lemma 2.13 Let l(f) = log λ(f). Then l : Bir(Pk ) → [0, +∞) is a length function. 1 n n Proof. Without loss of generality, we assume that λ(f) = limn→∞ deg(f ) , while the other case can be considered similarly. For any k ∈ N, it is easy that l(f k) = nk log deg f Pn limn→∞ n = kl(f). For any h ∈ Bir( k ), we have log deg hf nh−1 log deg f nk l(hfh−1) = lim = lim = l(f). n→∞ n n→∞ n For commuting maps f, g, we have (fg)n = f ngn. Therefore, log deg f ngn l(fg) = lim n→∞ n log deg f n log deg gn ≤ lim + lim = l(f)+ l(g). n→∞ n n→∞ n This checks the three conditions of the length function. It is surprising that when n = 2 and k is an algebraically closed field, the length function l(f) is given by the translation length τ(f) on an (infinite-dimensional) Gromov δ-hyperbolic space (see Blanc-Cantat [8], Theorem 4.4). Some other length functions are studied by Blanc and Furter [9] for groups of birational maps, eg. dynamical number of base-points and dynamical length.

3 Groups with purely positive length functions

Definition 3.1 A length function l on a group G is said to be purely positive if l(g) > 0 for any infinite-order element g ∈ G. In this section, we show that the (Gromov) hyperbolic group, mapping class group and outer automorphism groups of free groups have purely positive length functions. First, let us recall the relevant definitions. A geodesic metric space X is δ-hyperbolic (for some real number δ > 0) if for any geodesic triangle ∆xyz in X, one side is contained the δ-neighborhood of the other two sides. A group G is (Gromov) hyperbolic if G acts properly discontinuously and cocompactly on a δ-hyperbolic space X.

15 Deﬁnition 3.2 (i) An element g in a group G is called primitive if it cannot be writen as a proper power αn, where α ∈ G and |n| ≥ 2;

(ii) A group G has unique-root property if every inﬁnite-order element g is a proper n m power of a unique (up to sign) primitive element, i.e. g = γ = γ1 for primitive ± elements γ,γ1 will imply γ = γ .

The following fact is well-known.

Lemma 3.3 A torsion-free hyperbolic group has unique-root property.

Proof. Let G be a torsion-free hyperbolic group and 1 =6 g ∈ G. Suppose that g = γn = m γ1 for primitive elements γ and γ1. The set CG(g) of centralizers is virtually cyclic (cf. [16], Corollary 3.10, page 462). By a result of Serre, a torsion-free virtually free group is free. Since G is torsion-free, the group CG(g) is thus free and thus cyclic, say generated by t. Since γ and γ′ are primitive, they are t±.

Remark 3.4 The previous lemma does not hold for general hyperbolic groups with tor- sions. For example, let G = Z/2 × Z. We have (0, 2) = (0, 1)2 = (1, 1)2 and (0, 1), (1, 1) are both primitive.

For a group G, let P (G) be the set of all primitive elements. We call two primitive elements γ,γ′ are general conjugate if there exists g ∈ G such that gγg−1 = γ′ or gγ−1g = ′ γ . Let CP(G) be the general conjugacy classes of primitive elements. For a set S, let SR be the set of all real functions on S. The convex polyhedral cone spanned by S is the subset

{ s∈S ass | as ≥ 0 } ⊂ SR. P Lemma 3.5 Let G be a torsion-free hyperbolic group. The set of all length functions on G is the convex polyhedral cone spanned by the general conjugacy classes CP(G).

Proof. Let l be a length function on G. Then l gives an element s∈S ass in the convex polyhedral cone by as = l(s). Conversely, for any general conjugacyP classes [s] ∈ CP(G) ± with s a primitive element, let ls be the function defined by ls(s ) = 1 and ls(γ)=0 for element γ in any other general conjugacy classes. For any 1 =6 g ∈ G, there is a unique n (up to sign) primitive element γ such that g = γ . Define ls(g)= |n|ls(γ). Then ls satisfies conditions (1) and (2) in Definition 1.1. The condition (3) is satisfied automatically, since any commuting pair of elements a, b generate a cyclic group in a torsion-free hyperbolic group. Any element s∈S ass gives a length function on G as a combination of asls. P Lemma 3.6 Let G be one of the following groups:

• automorphism group Aut(Fk) of a free group;

• outer automorphism group Out(Fk) of a free group or

• mapping class group MCG(Σg,m) (where Σg,m is an oriented surface of genus g and m punctures);

• a hyperbolic group,

• a CAT(0) group or more generally

16 • a semi-hyperbolic group,

• a group acting properly semi-simply on a CAT(0) space,

• a group acting properly semi-simply on a δ-hyperbolic space.

Then G has a purely positive length function.

Proof. Note that hyperbolic groups and CAT(0) groups are semihyperbolic (see [16], Prop. 4.6 and Cor. 4.8, Chapter III.Γ). For a semihyperbolic group G acting a metric space X (actually X = G), the translation τ is a length function by Lemma 2.7. Moreover, for any infinite-order element g ∈ G, the length τ(g) > 0 (cf. [16], Lemma 4.18, page 479). For group acting properly semisimply on a CAT(0) space (or a δ-hyperbolic space), the translation d(x, γnx) l(γ) = lim n→∞ n is a length function (cf. Lemma 2.7). For any hyperbolic γ, we get l(γ) > 0. For any elliptic γ, it is finite-order since the action is proper. Alibegovic [1] proves that the stable word length of Aut(Fn), Out(Fn) are purely positive. Farb, Lubotzky and Minsky [25] prove that Dehn twists and more generally all elements of infinite order in MCG(Σg,m) have positive translation length.

Deﬁnition 3.7 A group G is called poly-positive (or has a poly-positive length), if there is a subnormal series 1= Hn ⊳ Hn−1 ⊳ ··· ⊳ H0 = G such that every ﬁnitely generated subgroup of the quotient Hi/Hi+1 (i = 0, ..., n − 1) has a purely positive length function.

Recall that a group G is poly-free, if there is a subnormal series 1 = Hn ⊳ Hn−1 ⊳ ··· ⊳ H0 = G such that the successive quotient Hi/Hi+1 is free (i = 0, ..., n − 1). Since a free group is hyperbolic, it has a purely positive length function. This implies that a poly-free group is poly-positive. A group is said to have a virtual property if there is a finite-index subgroup has the property. Let Σ be a closed oriented surfaec endowed with an area form ω. Denote by Diff(Σ,ω) the group of diffeomorphisms preserving ω and Diff0(Σ,ω) the subgroup consisting of diffeomorphisms isotopic to the identity.

Lemma 3.8 When the genuse of Σ is greater than 1, the group Diﬀ0(Σ,ω) and Diﬀ(Σ,ω) is poly-positive.

Proof. This is eseentially proved by Py [54] (Section 1). There is a group homomorphism

α : Diff0(Σ,ω) → H1(Σ, R) with ker α = Ham(Σ,ω) the group of Hamiltonian diffeomorphisms of Σ. Polterovich [53] (1.6.C.) proves that any finitely generated group of Ham(Σ,ω) has a purely positive stable word length. Since the quotient group Diff(Σ,ω)/Diff0(Σ,ω) is a subgroup of the mapping class group MCG(Σ), which has a purely positive stable word length by Farb-Lubotzky- Minsky [25], the group Diff(Σ,ω) is poly-positive.

17 4 Vanishing of length functions on abelian-by-cyclic groups

We will need the following result proved in [28].

Lemma 4.1 Given a group G, let l : G → [0, +∞) be function such that 1) l(e) = 0; 2) l(xn)= |n|l(x) for any x ∈ G, any n ∈ Z; 3) l(xy) ≤ l(x)+ l(y) for any x, y ∈ G. Then there exist a real Banach space (B, kk) and a group homomorphism ϕ : G → B such that l(x) = kϕ(x)k for all x ∈ G. Further more, if l(x) > 0 for any x =6 e, one can take ϕ to be injective, i.e., an isometric embedding.

a b Let Z2 ⋊ Z be an abelian by cyclic group, where A = ∈ GL (Z). We prove A c d 2 Theorem 0.1 by proving the following two theorems.

Theorem 4.2 When the absolute value of the trace |tr(A)| > 2, any length function 2 2 l : Z ⋊A Z → R≥0 vanishes on Z .

a b Proof. Let A = ∈ GL (Z). Suppose that t is a generator of Z and c d 2

x x t t−1 = A y y for any x, y ∈ Z. Note that

(0, tk)(v, 0)(0, tk)−1 =(Akv, 0) for any v ∈ Z2 and k ∈ Z. Therefore, an element v ∈ Z2 is conjugate to Akv for any integer k. Note that 1 a 1 a2 + bc A = , A2 = 0 c 0 ac + dc and a2 + bc a 1 =(a + d) − (ad − bc) . ac + dc c 0 Therefore, we have 1 a |a + d|l( ) = l((a + d) ) 0 c a2 + bc 1 = l( +(ad − bc) ) ac + dc 0 1 ≤ (1 + |ad − bc|)l( ). 0

1 When ad − bc = ±1 and |a + d| > 2, we must have l( )=0. Similarly, we can prove 0 0 that l( )=0. Since l is subadditive on Z2, we get that l vanishes on Z2. 1

18 Theorem 4.3 When the absolute value |tr(A)| = |a + d| = 2,I2 =6 A ∈ SL2(Z), any 2 2 length function l : Z ⋊A Z → R≥0 vanishes on the direct summand of Z spanned by eigenvectors of A.

1 n Proof. We may assume that A = , n =6 0. For any integer k ≥ 0 and v ∈ Z2, we 0 1 have tkvt−k = Akv. 0 Take v = to get that 1 0 kn 0 tk t−k = + . 1 0 1

Since the function l|Z2 is given by the norm of a Banach space according to Lemma 4.1, we get that

1 0 0 k|n|l( ) ≤ l(tk t−k)+ l( ) 0 1 1 0 = 2l( ). 1

1 Since k is arbitrary, we get that l( )=0. 0

1 1 Remark 4.4 When A = , the semidirect product G = Z2 ⋊ Z is a Heisenberg 0 1 A group. A length function on G/Z(G) ∼= Z2 gives a length function on G. In particular, a 0 length function of G may not vanish on the second component ∈ Z2 < G. 1

Remark 4.5 When A ∈ SL2(Z) has |tr(A)| < 2, the matrix A is of finite order and 2 3 the semi-direct product Z ⋊A Z contains Z as a finite-index normal subgroup. Actually, 2 in this case the group Z ⋊A Z is the fundamental group of a flat 3-manifold M (see 2 [61], Theorem 3.5.5). Therefore, the group Z ⋊A Z acts freely properly discontinuously isometrically and cocompactly on the universal cover M˜ = R3. This means the translation 2 length gives a purely positive length function on Z ⋊A Z.

n Lemma 4.6 Let A ∈ GLn(Z) be a matrix and G = Z ⋊A Z the semi-direct product. Let n i k i=0 aix be the characterisitic polynomial of some power A . Suppose that for some k, thereP is a coeﬃcient ai such that |ai| > |aj|. Xj6=i Any length function l of G vanishes on Zn.

Proof. Let t be a generator of Z and tat−1 = Aa for any a ∈ Zn. Note that for any integer m, we have tmat−m = Ama and l(a)= l(Ama). Note that

n ki aiA =0 Xi=0

19 and thus n ki aiA a = 0, Xi=0 n ki l( aiA a) = 0 Xi=0 for any a ∈ Zn. Therefore,

ki kj |ai|l(a)= |ai|l(A a)= l( ajA a) ≤ |aj|l(a). Xj6=i Xj6=i This implies that l(a)=0.

Proof of Corollary 0.2. When the group action is C2, define log sup kD f nk log sup kD f −nk l(f) = max{ lim x∈M x , lim x∈M x } n→+∞ n n→+∞ n for any diffeomorphism f : M → M. Lemma 2.5 shows that l is a length function, which is an upper bound of the Lyapunov exponents. When the group action is Lipschitz, define d(fx,fy) L(f) = sup x6=y d(x, y) for a Lipschitz-homeomorphism f : M → M. Since L(fg) ≤ L(f)L(g) for two Lipschitz- max{log(L(f n)),log(L(f −n))} homeomorphisms f,g : M → M, we have that l(f) := limn→∞ n gives a length function by Lemma 2.1. Note that l(f) ≥ htop(f) (see [40], Theorem 3.2.9, page 124). The vanishings of the topological entropy htop and the Lyapunov exponents in Corolary 0.2 are proved by Theorem 0.1 considering these length functions.

5 Classiﬁcation of length functions on nilpotent groups

The following lemma is a key step for our proof of the vanishing of length functions on Heisenberg groups.

Lemma 5.1 Let G = ha, b, c | aba−1b−1 = c,ac = ca, bc = cbi be the Heisenberg group. Suppose that f : G → R is a conjugation-invariant function, i.e. f(xgx−1)= f(g) for any x, g ∈ G. For any coprime integers (not-all-zero) m, n and any integer k, we have f(ambnck)= f(ambn).

Proof. It is well-known that for any integers n, m, we have [an, bm] = cnm. Actually, since aba−1b−1 = c, we have ba−1b−1 = a−1c and thus ba−nb−1 = a−ncn for any integer n. Therefore, anba−nb−1 = cn and anba−n = cnb, anbma−n = anmbm for any integer m. This means [an, bm] = cnm. For any coprime m, n, and any integer k, let s, t ∈ Z such that ms + nt = k. We have

a−mb−sambs = cms, b−sambs = amcms, b−sambnbs = ambncms

20 and atbna−tb−n = cnt, atbna−t = bncnt, atambna−t = ambncnt. Therefore,

at(b−sambnbs)a−t = at(ambncms)a−t = ambncnt+ms = ambnck.

When f is conjugation-invariant, we get f(ambnck)= f(ambn) for any coprime m, n, and any integer k.

Lemma 5.2 Let G = ha, b, c | aba−1b−1 = c,ac = ca, bc = cbi be the Heisenberg group. Any length function l : G → [0, ∞) (in the sense of Deﬁnition ∼ 2 1.1) factors through the abelization Gab := G/[G,G] = Z . In other words, there is a function ′ l : Gab → [0, ∞) ′ n ′ ′ such that l (x )= |n|l (x) for any x ∈ Gab, any integer n and l = l ◦q, where q : G → Gab is the natural quotient group homomorphism.

Proof. Let H = hci ∼= Z and write G = ∪gH the union of left cosets. We choose the i j 2 representative gij = a b with (i, j) ∈ Z . Note that the subgroup hgij,ci generated by 2 gij,c is isomorphic to Z for coprime i, j. The length function l is subadditive on hgij,ci. By Lemma 4.1, there is a Banach space B and a group homomorphism ϕ : hgij,ci → B such that l(g)= kϕ(g)k for any g ∈ hgij,ci. Lemma 5.1 implies that

k kϕ(gij)+ ϕ(c )k = kϕ(gij)k for any integer k. Since

k kϕ(gij)k = kϕ(gij)+ ϕ(c )k≥|k|kϕ(c)k−kϕ(gij)k for any k, we have that kϕ(c)k = l(c)=0. This implies that

n m n m n l(gijc )= kϕ(gij)+ ϕ(c )k = l(gij) for any integers m, n. Moreover, for any integers m, n and coprime i, j, we have anibnjcm = (aibj)nck for some integer k. This implies that

l(anibnjcm)= l((aibj)n)= |n|l(aibj).

Therefore, the function l is constant on each coset gH. Define l′(gH)= l(g). Since l(gk)= |l|l(g), we have that l′(gkH)= |k|l′(gH). The proof is finished. Denote by S′ = {(m, n) | m, n are coprime integers} be the set of coprime integer pairs and define an equivalence relation by (m, n) ∼ (m′, n′) if (m, n) = ±(m′, n′). Let S = S′/ ∼ be the equivalence classes.

Theorem 5.3 Let G = ha, b, c | aba−1b−1 = c,ac = ca, bc = cbi be the Heisenberg group. The set of all length functions l : G → [0, ∞) (in the sense of Deﬁnition 1.1) is the convex polyhedral cone

R≥0[S]= { ass | as ∈ R≥0,s ∈ S}. Xs∈S

21 Proof. Similar to the proof of the previous lemma, we let H = hci ∼= Z and write i j 2 G = ∪gH the union of left cosets. We choose the representative gij = a b with (i, j) ∈ Z . Let T be the set of length function l : G → [0, ∞). For any length functions l, let i j ϕ(l)= s∈S ass ∈ R≥0[S], where as = l(a b ) with (i, j) a representative of s. Note that P l(a−ib−j)= l(b−ja−icij)= l(b−ja−i)= l(aibj), which implies that as is well-deﬁned. We have deﬁned a function ϕ : T → R≥0[S]. If i j i j ϕ(l1) = ϕ(l2) for two functions l1,l2, then l1(a b ) = l2(a b ) for coprime integers i, j. Since both l1,l2 are conjugation-invariant, Lemma 5.1 implies that l1,l2 coincide on any coset aibjH and thus on the whole group G. This proves the injectivity of ϕ. For any

s∈S ass, we deﬁne a function P l : G = ∪aibjH → [0, ∞).

i j For any coprime integers i, j, define l(a b z) = as for any representative (i, j) of s and any z ∈ H. For any general integers m, n and z ∈ H, define l(ambnz) = l(ambn) = | gcd(m, n)|l(am/ gcd(m,n)bn/ gcd(m,n)) and l(z)=0. From the definition, it is obvious that l is homogeous. Note that any element of G is of the form akbsct for integers k,s,t ∈ Z. ′ ′ ′ For any two elements akbsct, ak bs ct we have the conjugation

′ ′ ′ ′ ′ ′ ′′ ak bs ct akbsct(ak bs ct )−1 = akbsct for some t′′ ∈ Z. Therefore, we see that l is conjugation-invariant. The previous equality also shows that two elements g, h are commuting if and only if they lies simutanously in haibj,ci for a pair of coprime integers i, j. By construction, we have l(g) = l(h). This proves the surjectivity of ϕ.

6 Length functions on matrix groups

In this section, we study length functions on matrix groups SLn(R). As the proofs are elementary, we present here in a separated section, without using profound results on Lie groups and algebraic groups. The following lemma is obvious.

p −1 q Lemma 6.1 Let Gp,q = hx, t : tx t = x i be a Baumslag-Solitar group. When |p|= 6 |q|, any length function l on G has l(x)=0.

Proof. Note that |p|l(x)= l(xp)= l(xq)= |q|l(x), which implies l(x)=0. Let V n be a ﬁnite-dimensional vector space over a ﬁeld K and A : V → V a unipotent linear transformation (i.e. Ak = 0 for some positive integer k). The following fact is from linear algebra (see the Lemma of page 313 in [4]. Since the reference is in Chinese, we repeat the proof here).

Lemma 6.2 I+A is conjugate to a direct sum of Jordan blocks with 1s along the diagonal.

Proof. We prove that V has a basis

k1−1 k2−1 ks−1 {a1, Aa1, ··· , A a1, a2, Aa2, ··· , A a2, ··· , as, ··· , Aas, ··· , A as}

ki satisfying A ai = 0 for each i, which implies that the representation matrix of I + A is similar to a direct sum of Jordan blocks with 1 along the diagonal. The proof is based

22 on the induction of dim V. When dim V = 1, choose 0 =6 v ∈ V. Suppose that Av = λv. Then Akv = λkv = 0 and thus λ = 0. Suppose that the case is proved for vector spaces of dimension k < n. Note that the invariant subspace AV is a proper subspace of V (otherwise, AV = V implies AkV = Ak−1V = V = 0). By induction, the subspace AV has a basis

k1−1 k2−1 ks−1 {a1, Aa1, ··· , A a1, a2, Aa2, ··· , A a2, ··· , as, ··· , Aas, ··· , A as}.

Choose bi ∈ V satisfying A(bi)= ai. Then A maps the set

k1 k2 ks {b1, Ab1, ··· , A b1, b2, Ab2, ··· , A b2, bs, ··· , Abs, ··· , A bs} to the basis

k1−1 k2−1 ks−1 {a1, Aa1, ··· , A a1, a2, Aa2, ··· , A a2, ··· , as, ··· , Aas, ··· , A as}.

ki This implies that the former set is linearly independent (noting that A(A bi) = 0). Extend this set to be a V ′s basis

k1−1 k2−1 ks−1 {b1, Ab1, ··· , A b1, b2, Ab2, ··· , A b2, bs, ··· , Abs, ··· , A bs, bs+1, ··· , bs′ }.

ki+1 ki Note that Abi = 0 for i ≥ s + 1 and A bi = A ai = 0 for each i ≤ s. This ﬁnishes the proof.

Corollary 6.3 Let An×n be a strictly upper triangular matrix over a ﬁeld K of characteristic ch(K) =26 . Then A2 is conjugate to A.

Proof. Suppose that A = I + u for a nilpotent matrix u. Lemma 6.2 implies that A2 is conjugate to a direct sum of Jordan blocks. Without loss of generality, we assume A is a Jordan block. Then A2 = I +2u + u2. By Lemma 6.2 again, A2 is conjugate to a direct sum of Jordan blocks with 1s along the diagonal. The minimal polynomial of A2 is (x − 1)n, which shows that there is only one block in the direct sum and thus A2 is conjugate to A. Recall that a matrix A ∈ GLn(R) is called semisimple if as a complex matrix A is conjugate to a diagonal matrix. A semisimple matrix A is elliptic (respectively, hyperbolic) if all its (complex) eigenvalues have modulus 1 (respectively, are > 0). The following lemma is the complete multiplicative Jordan (or Jordan-Chevalley) decomposition (cf. [34], Lemma 7.1, page 430).

Lemma 6.4 Each A ∈ GLn(R) can be uniquely writen as A = ehu, where e, h, u ∈ GLn(R) are elliptic, hyperbolic and unipotent, respectively, and all three commute.

The following result characterizes the continuous length functions on compact Lie groups.

Lemma 6.5 Let G be a compact connected Lie group and l a continuous length function on G. Then l =0.

Proof. For any element g ∈ G, there is a maximal torus T g. For ﬁnite order element h ∈ T, we have l(h)=0. Note that the set of ﬁnite-order elements is dense in T. Since l is continuous, l vanishes on T and thus l(g)= 0 for any g.

23 Theorem 6.6 Let G = SLn(R) (n ≥ 2). Let l : G → [0, +∞) be a length function, which is continuous on compact subgroups and the subgroup of diagonal matrices with positive diagonal entries. Then l is uniquely determined by its images on the subgroup D of diagonal matrices with positive diagonal entries.

Proof. For any g ∈ SLn(R), let g = ehu be the Jordan decomposition for commuting elements e, h, u, where e is elliptic, h is hyperbolic and u is unipotent (see Lemma 6.4) after multiplications by suitable powers of derterminants. Then

l(g) ≤ l(e)+ l(h)+ l(u).

For any unipotent matrix u, there is an invertible matrix a such that aua−1 is strictly upper triangular (see [34], Theorem 7.2, page 431). Lemma 6.3 implies that u2 is conjugate to u. Therefore, l(u) = 0 by Lemma 6.1. Since l vainishes on a compact Lie group (cf. Lemma 6.5), we have that l(e) = 0 for any elliptic matrix e. Therefore, l(g) ≤ l(h). Similarly, l(h)= l(e−1gu−1) ≤ l(g) which implies l(g) = l(h). Note that a hyperbolic matrix is conjugate to a real diagonal matrix with positive diagonal entries.

Proof of Theorem 0.10. By Theorem 6.6, the length function l is determined by its t image on the subgroup D generated by h12(x), x ∈ R>0. Take x = e , t ∈ R. We have k l |k| k l(h12(e )) = |l| l(h12(e)) for any rational number l . Since l satisﬁes the condition 2) of the deﬁnition and is continuous on D, we see that l|D is determined by the image l(h12(e)) (actually, any real number t is a limit of a rational sequence). Note that the translation function τ vainishes on compact subgroups and is continuous on the subgroup of diagonal subgroups with positive diagonal entries (cf. [16], Cor. 10.42 and Ex. 10.43, page 320). Therefore, l is proposional to τ. Actually, τ can be determined explicitly by the formula τ (A) tr(A) = ±2cosh 2 (for nonzero τ(A)), where tr is the trace and cosh is the hyperbolic cosine function (see [5], Section 7.34, page 173). This implies that l(A) is determined by the spectrum radius of A (which could also be seen clearly by the matrix norm).

Let h1i(x) (i = 2, ··· , n) be an n × n diagonal matrix whose (1, 1)-entry is x, (i, i)- entry is x−1, while other diagonal entries are 1s and non-diagonal entries are 0s. The subgroup D < SLn(R) of diagonal matrices with positive diagonal entries is isomorphic n−1 to (R>0) and D is generated by the matrices h1i(x) (i = 2, ··· , n) whose (1, 1)-entry −1 is x, (i, i)-entry is x . Since h1i(x) (i =6 1) is conjugate to h12(x), a length function l : SLn(R) → [0, +∞) is completely determined by its image on the convex hull spanned by h12(e), h13(e), ··· , h1n(e) (see Theorem 7.10 for a more general result on Lie groups). Here e is the Euler’s number in the natural exponential function.

Corollary 6.7 Let l : SL2(R) → [0, +∞) be a non-trivial length function that is continuous on the subgroup SO(2) and the diagonal subgroup. Then l(g) > 0 if and only if g is hyperbolic.

Proof. It is well-known that the elements in SL2(R) are classiﬁed as elliptic, hyperbolic and parabolic elements. Moreover, the translation length τ vanishes on the compact subgroup SO(2) and the parabolic elements. The corollary follows Theorem 0.10. When the length function l is the asymptotic distortion function dist∞, Corollary 6.7 is known to Navas [24] (Proposition 4).

24 7 Length functions on algebraic and Lie groups

For an algebraic group G, let k[G] be the regular ring. For any g ∈ G, let ρg : k[G] → k[G] be the right translation by x. The following is the famous Jordan (or Jordan-Chevalley) decomposition.

Lemma 7.1 ([37] p.99) Let G be an algebraic group and g ∈ G. There exists unique elements gs,gu such that gsgu = gugs, and ρgs is semisimple, ρgu is unipotent.

Lemma 7.2 [46] Let G be a reductive connected algebraic group over an algebraically closed ﬁeld k. The conjugacy classes of unipotent elements in G is ﬁnite.

Lemma 7.3 ([29], Theorem 3.4) If G is a reductive linear algebraic group defined over a field k and g ∈ G(k) then the set of conjugacy classes in G(k) which when base changed to the algebraic closed field k¯ are equal to the conjugacy class of g in G(k¯) is in bijection with the subset of H1(k/k,Z¯ (g)(k)), the Galois cohomology group.

Definition 7.4 A field k is of type (F) if for any integer n there exist only finitely many extensions of k of degree n (in a fixed algebraic closure k¯ of k).

Examples of fields of type (F) include: the field R of reals, a finite field, the field of formal power series over an algebraically closed field.

Lemma 7.5 [Borel-Serre [12], Theorem 6.2] Let k be a field of type (F) and let H be a linear algebraic group defined over k. The set H1(k/k,H¯ (k)) is finite.

Lemma 7.6 Let G(k) be a reductive linear algebraic group over a ﬁeld of type (F) and l a length function on G. Then l(g)= l(gs), where gs is the semisimple part of g.

Proof. By the Jordan decomposition g = gsgu, we have l(g) ≤ l(gs)+ l(gu) and l(gs) ≤ −1 n l(g) + l(gu ). Note that for any integer n, gu is also unipotent. By the Lemma 7.2, Lemma 7.3 and Lemma 7.5, there are only ﬁnitely many conjugacy classes of unipotent n1 n2 elements. This implies that gu = gu for distinct positive integers n1, n2. Therefore, we have n1l(gu)= n2l(gu), which implies that l(gu) = 0 and thus l(g)= l(gs).

A Lie group G is semisimple if its maximal connected solvable normal subgroup is trivial. Let g be its Lie algebra and let exp : g → G denote the exponential map. An element x ∈ g is real semi-simple if Ad(x) is diagonalizable over R. An element g ∈ G is called hyperbolic (resp. unipotent) if g is of the form g = exp(x) where x is real semisimple (resp. nilpotent). In either case the element x is easily seen to be unique and we write x = log g. The following is the Jordan decomposition in Lie groups. An element e ∈ G is elliptic if Ad(e) is diagonalizable over C with eigenvalues 1.

Lemma 7.7 ([42], Prop. 2.1 and Remark 2.1)

1. Let g ∈ G be arbitrary. Then g may be uniquely written

g = e(g)h(g)u(g)

where e(g) is elliptic, h(g) is hyperbolic and u(g) is unipotent and where the three elements e(g), h(g),u(g) commute.

25 2. An element f ∈ G commutes with g if and only if f commutes with the three components. Moreover, if f,g commutes, then e(fg)= e(f)e(g), h(fg)= h(f)h(g),u(fg)= u(f)u(g).

Lemma 7.8 [Eberlein, Prop. 1.14.6, page 63]Let G be a connected semisimple Lie group whose center is trivial. Then there exists an integer n ≥ 2 and an algebraic group G∗ < ∗0 GLn(C) deﬁned over Q such that G is isomorphic to GR (the connected component of ∗0 GR containing the identity) as a Lie group.

Let G = KAN be an Iwasawa decomposition. The Weyl group W is the ﬁnite group deﬁned as the quotient of the normalizer of A in K modulo the centralizer of A in K. For an element x ∈ A, let W (h) be the set of all elements in A which are conjugate to x in G.

Lemma 7.9 ([42], Prop. 2.4) An element h ∈ G is hyperbolic if and only if it is conjugate to an element in A. In such a case, W (h) is a single W -orbit in A.

Theorem 7.10 Let G be a connected semisimple Lie group whose center is finite with an Iwasawa decomposition G = KAN. Let W be the Weyl group. (i) Any length function l on G that is continuous on the maximal compact subgroup K is determined by its image on A. (ii) Conversely, any length function l on A that is W -invariant (i.e. l(w · a) = l(a)) can be extended to be a length function on G that vanishes on the maximal compact subgroup K. Proof. (i) Let Z be the center of G. Then G/Z is connected with trivial center. For any z ∈ Z,g ∈ G, we have l(z) = 0 and l(gz)= l(g). The length function l factors through a length function on G/Z. We may assume that G has the trivial center. For any g ∈ G, the Jordan decomposition gives g = ehu, where e is elliptic, h is hyperbolic and u is unipotent and where the three elements e, h, u commute (cf. Lemma 7.7). By Lemma 7.8, the Lie group G is an algebraic group. Lemma 7.6 implies that l vanies on unipotent elements and l(g)= l(eh). Since l vanishes on e (cf. Lemma 6.5), we have l(g) = l(h). Therefore, the function l is determined by its image on A. (ii) Let l be a length function l on A that is W -invariant. We first extend l to the set H of all the conjugates of A. For any g ∈ G, a ∈ A, define l′(gag−1)= l(a). If

−1 −1 g1a1g1 = g2a2g2 −1 −1 for g1,g2 ∈ G, a1, a2 ∈ A, then g1 g2a2g2 g1 = a1. By Lemma 7.9, there exists an element ′ w ∈ W such that w·a2 = a1. Therefore, we have l(a1)= l(a2) and thus l is well-deﬁned on the set H of conjugates of elements in A. Such a set H is the set consisting of hyperbolic elements by Lemma 7.9. We then extend l′ on the set of all conjugates of elements in K. For any g ∈ G,k ∈ K, deﬁne l′(gkg−1)=0. If

−1 −1 g1kg1 = g2ag2 −1 −1 for g1,g2 ∈ G,k ∈ K, a ∈ A, then g1 g2ag2 g1 = k. Then k is both hyperbolic and elliptic. The only element which is both elliptic and hyperbolic is the identity element. Therefore, we have k = a = 1 and ′ −1 −1 l (g1kg1 )= l(g2ag2 )=1.

26 This shows that l′ is well-defined on the set of hyperbolic elements and elliptic elements. For any unipotent element u ∈ G, define l′(u)=0. For any element g ∈ G, let g = ehu be the Jordan decomposition. Define l′(g)= l′(h). We check the function l′ is a length function on G. The definition shows that l′ is conjugate invariant. For any positive integer n and any g ∈ G with Jordan decomposition g = ehu, we have gn = enhnun and thus l′(gn) = l′(hn). But hn is hyperbolic and conjugate to an element in A (see Lemma 7.9). Therefore, we have l′(hn) = |n|l′(h) and ′ n ′ thus l (g )= |n|l (g). If g1 = e1h1u1 commutes with g2 = e2h2u2, then

g1g2 = e1e2h1h2u1u2

′ ′ (cf. Lemma 7.7) and l (g1g2)= l (h1h2). Since h1, h2 are commuting hyperbolic elements, they are conjuate simutaniously to elements in A. Therefore, we have

′ ′ ′ l (h1h2) ≤ l (h1)+ l (h2) and ′ ′ ′ l (g1g2) ≤ l (g1)+ l (g2).

Remark 7.11 A length function l on A is determined by a group homomorphism f : A → B, for a real Banach space (B, kk), satisfying l(a) = kf(a)k (see Lemma 4.1). The previous theorem implies that a length function l on the Lie group G (that is continuous on compact subgroup) is uniqely determined by such a group homomorphism f : A → B such that kf(a)k = kf(wa)k for any a ∈ A and w ∈ W, the Weyl group. Let G be a connected semisimple Lie group whose center is finite with an Iwasawa decomposition G = KAN. Let exp : g → G be the exponent map from the Lie algebra g with subalgebra h corresponding to A. Theorem 7.12 Suppose that l is a length function on G that is continuous on K and A. Then l is determined by its image on exp(v) (unit vector v ∈ h) in a fixed closed Weyl chamber of A. Proof. Let Z be the center of G. Then G/Z is connected with trivial center. The length function l factors through an length function on G/Z (cf. Corrolary 1.8). For any g ∈ G we have g = ehu, where e is elliptic, h is hyperbolic and u is unipotent and where the three elements e, h, u commute (cf. Lemma 7.7). By Lemma 7.8 and Lemma 7.6, we have l(g) = l(eh). Since l vanishes on e (cf. Lemma 6.5), we have l(g) = l(h). Any element h ∈ A is conjugate to an element in a fixed Weyl chamber C (cf. [38], Theorem 8.20, page 254). For any element exp(x) ∈ C, with unit vector x ∈ h, the one-parameter subgroup exp(Rx) lies in A. Since l is continuous on A, the function l is determined by its image on m n x 1 mx m x m exp(Qx). Note that l(e )= n l(e )= n l(e ) for any rational number n . The function l is determined by l(ex), for all unit vectors x in the fixed closed Weyl chamber. Corollary 7.13 Let G be a connected semisimple Lie group whose center is finite of real rank 1. There is essentially only one length function on G. In order words, any continuous length function is proportional to the translation function on the symmetric space G/K. Proof. When the real rank of G is 1, a closed Weyl chamber is of dimension 1. Therefore, the previous theorem implies that any continuous length function is determined by its image on a unit vector in a split torus.

27 8 Rigidity of group homomorphisms on arithmetic groups

Let V denote a ﬁnite-dimensional vector space over C, endowed with a Q-structure. Recall that the arithmetic subgroup is deﬁned as the following (cf. Borel [11], page 37).

Deﬁnition 8.1 Let G be a Q-subgroup of GL(V ). A subgroup Γ of GQ is said to be arithmetic if there exists a lattice L of VQ such that Γ is commensurable with GL = {g ∈ GQ : gL = L}.

Theorem 8.2 Let Γ be an arithmetic subgroup of a simple algebraic Q-group of Q-rank at least 2. Suppose that H is a group with a purely positive length function. Then any group homomorphism f :Γ → H has its image ﬁnite.

Recall that a group G is quasi-simple, if any non-trivial normal subgroup is either ﬁnite or of ﬁnite index. The Margulis-Kazhdan theorem (see [62], Theorem 8.1.2) implies that an irreducible lattice (and hence) in a semisimple Lie group of real rank ≥ 2 is quasi-simple.

Lemma 8.3 Let Γ be a ﬁnitely generated quasi-simple group with contains a Heisen- berg subgroup, i.e. there are elements torsion-free elements a, b, c ∈ Γ satisfying [a, b] = c, [a, c] = [c, b]=1. Suppose that G has a virtually poly-positive length. Then any group homomorphism f :Γ → G has its image ﬁnite.

Proof. Suppose that G has a ﬁnite-index subgroup H and a subnormal series

1= Hn ⊳ Hn−1 ⊳ ··· ⊳ H0 = H such that every ﬁnitely generated subgroup of Hi/Hi+1 has a purely positive length function. Without loss of generality, we assume that H is normal. Let f : Γ → G be a homomorphism. The kernel of the composite

f f0 :Γ → G → G/H is ﬁnitely generated. Suppose that the image of the composite

f f1 : ker f0 → H → H/H1 has a purely positive length function l. After passing to finite-index subgroups, we may still suppose that ker f0 contains a Heisenberg subgroup ha, b, ci. By Lemma 5.2, the length k function l vanishes on f1(c). Therefore f1(c )=1 ∈ H/H1 for some integer k > 0. The k normal subgroup ker f1 containing c is of finite index. Now we have map ker f1 → H1 induced by f. An induction argument shows that f maps some power cd of the central element of the Heisenberg subgroup into the identity 1 ∈ G. Therefore, the image of f is finite. Proof of Theorem 8.2. It is well-known that that G contains a Q-split simple subgroup whose root system is the reduced subsystem of the Q-root system of G (see [13], Theorem 7.2, page 117). Replacing G with this Q-subgroup, we may assume G is Q-split and the root system of G is reduced. Because G is simple and Q-rank(G)≥ 2, we know that the Q-root system of G is irreducible and has rank at least two. Therefore, the Q-root

28 system of G contains an irreducible subsystem of rank two, that is, a root subsystem of type A2, B2,G2 (see [60], page 338). For A2, choose {α1,α3} as a set of simple roots (see Figure 1).

Then the root element xα1+α3 (rs)= xα2 (rs) is a commutator [xα1 (r), xα3 (s)], with xα2 (rs) commutes with xα1 (r), xα3 (s). For G2, the long roots form a subsystem of A2. For B2, choose {α1,α4} as a set of simple roots (see Figure 2). The long root element xα3 (2rs) is a commutator [xα2 (r), xα4 (s)] of the two short root elements, and xα3 (2rs) commutes with xα2 (r), xα4 (s) (cf. [37], Proposition of page 211). This shows that the arithmetic subgroup Γ contains a Heisenberg subgroup. The theorem is then proved by Lemma 8.3.

If we consider special length functions, general results can be proved. When we consider the stable word lengths, the following is essentially already known (cf. Polterovich [53], Corollary 1.1.D and its proof).

Proposition 8.4 Let Γ be an irreducible non-uniform lattice in a semisimple connected, Lie group without compact factors and with ﬁnite center of real rank ≥ 2. Assume that a group G has a virtually poly-positive stable word length. In other words, the group G has a ﬁnite-index subgroup H and a subnormal series

1= Hn ⊳ Hn−1 ⊳ ··· ⊳ H0 = H such that every ﬁnitely generated subgroup of Hi/Hi+1 (i = 0, 1, ··· , n − 1) has a purely positive stable word length. Then any group homomorphism f : Γ → G has its image ﬁnite.

Proof. Without loss of generality, we assume that f takes image in H. Since a lattice is ﬁnitely generated, Γ has its image in H0/H1 ﬁnitely generated. When the image has a purely positive word length, any distorted element in Γ must have trivial image in H0/H1 (see ). Lubotzky, Mozes and Raghunathan [44] prove that irreducible non-uniform lattices in higher rank Lie groups have non-trivial distortion elements (They actually prove the stronger result that there are elements in the group whose word length has logarithmic

29 growth). Then a ﬁnite-index subgroup Γ0 < Γ will have image in H1, since high-rank irreducible lattices are quasi-simple. An induction argument ﬁnishes the proof.

When we consider the length given by quasi-cocyles, the following is also essentially already known ( cf. Py [54], Prop. 2.2, following Burger-Monod [18] [19]). Recall that a locally compact group has property (TT) if any continuous rough action on a Hilbert space has bounded orbits (see [49], page 172). Burger-Monod proves that an irreducible lattice Γ in a high-rank semisimple Lie group has property (TT).

Proposition 8.5 Let Γ be an irreducible lattice in a semisimple connected, Lie group without compact factors and with ﬁnite center of real rank ≥ 2.. Assume that a group G has a virtually poly-positive average norm for quasi-cocycles. In other words, the group G has a ﬁnite-index subgroup H and a subnormal series

1= Hn ⊳ Hn−1 ⊳ ··· ⊳ H0 = H such that every ﬁnitely generated subgroup of Hi/Hi+1 (i = 0, 1, ··· , n − 1) has a purely positive length given by a quasi-cocycle with values in Hilbert spaces. Then any group homomorphism f :Γ → G has its image ﬁnite.

Proof. Note that a group Γ has property (TT) if and only if H1(Γ; E) = 0 and

2 2 ker(Hb (Γ; E) → H (Γ; E))=0

2 for any linear isometric action of Γ on a Hilbert space E. Here Hb (Γ; E) is the second bounded cohomology group. Suppose that u : Γ → E is a quasi-cocyle. There is a bounded map v :Γ → E and a 1-cocycle w :Γ → E such that

u = v + w, by Proposition 2.1 of Py [54]. Since Γ has property T, there exists x0 ∈ E such that w(γ)= γx0 − x0. Therefore, we have

ku(γn)k kv(γn)+ w(γn)k = n n kv(γn)+ γnx − x k = 0 0 n kv(γn)k +2kx k ≤ 0 → 0. n Without loss of generality, we assume that G = H. Suppose that any ﬁnitely generated subgroup of H/H1 has a purely positive average norm l given by a cocycle. The composite

f Γ → H → H/H1 has a finite-index kernel Γ0, since l vanishes on infinite-order elements of the image. This implies that f(Γ0) lies in H1. A similar argument proves that ker f is of finite index in the general case.

30 9 Rigidity of group homomorphisms on matrix groups

9.1 Steinberg groups over ﬁnite rings Recall that a ring R is right Artinian if any non-empty family of right ideals contains minimal elements. A ring R is semi-local if R/rad(R) is right Artinian (see Bass’ K- theory book [3] page 79 and page 86), where rad(R) is the Jacobson radical. Let n be a positive integer and Rn the free R-module of rank n with standard basis. A vector n (a1,...,an) in R is called right unimodular if there are elements b1,...,bn ∈ R such that a1b1 + ··· + anbn = 1. The stable range condition srm says that if (a1,...,am+1) is a right unimodular vector then there exist elements b1,...,bm ∈ R such that (a1 + am+1b1,...,am + am+1bm) is right unimodular. It follows easily that srm ⇒ srn for any n ≥ m. A semi-local ring has the stable range sr2 ( [3], page 267, the proof of Theorem 9.1). A ﬁnite ring R is right Artinian and thus has sr2. The stable range

sr(R) = min{m : R has srm+1}.

Thus sr(R) = 1 for a finite ring R. We briefly recall the definitions of the elementary subgroups En(R) of the general linear group GLn(R), and the Steinberg groups Stn(R). Let R be an associative ring with identity and n ≥ 2 be an integer. The general linear group GLn(R) is the group of all n × n invertible matrices with entries in R. For an element r ∈ R and any integers i, j such that 1 ≤ i =6 j ≤ n, denote by eij(r) the elementary n × n matrix with 1s in the diagonal positions and r in the (i, j)-th position and zeros elsewhere. The group En(R) is generated by all such eij(r), i.e.

En(R)= heij(r)|1 ≤ i =6 j ≤ n, r ∈ Ri.

−1 −1 Denote by In the identity matrix and by [a, b] the commutator aba b . The following lemma displays the commutator formulas for En(R) (cf. Lemma 9.4 in [47]).

Lemma 9.1 Let R be a ring and r, s ∈ R. Then for distinct integers i,j,k,l with 1 ≤ i,j,k,l ≤ n, the following hold:

(1) eij(r + s)= eij(r)eij(s);

(2) [eij(r), ejk(s)] = eik(rs);

(3) [eij(r), ekl(s)] = In.

By Lemma 9.1, the group En(R) (n ≥ 3) is finitely generated when the ring R is finitely generated. Moreover, when n ≥ 3, the group En(R) is normally generated by any elementary matrix eij(1). The commutator formulas can be used to define Steinberg groups as follows. For n ≥ 3, the Steinberg group Stn(R) is the group generated by the symbols {xij(r):1 ≤ i =6 j ≤ n, r ∈ R} subject to the following relations:

(St1) xij(r + s)= xij(r)xij(s);

(St2) [xij (r), xjk(s)] = xik(rs) for i =6 k;

31 (St3) [xij (r), xkl(s)]= 1 for i =6 l, j =6 k.

There is an obvious surjection Stn(R) → En(R) deﬁned by xij(r) 7−→ eij(r). For any ideal I ⊳ R, let p : R → R/I be the quotient map. Then the map p induces a group homomorphism p∗ : Stn(R) → Stn(R/I). Denote by Stn(R,I) (resp., En(R,I)) the subgroup of Stn(R) (resp., En(R)) normally generated by elements of the form xij(r) (resp., eij(r)) for r ∈ I and 1 ≤ i =6 j ≤ n. In fact, Stn(R,I) is the kernel of p∗ (cf. Lemma 13.18 in Magurn [47] and its proof). However, En(R,I) may not be the kernel of En(R) → En(R/I) induced by p.

Lemma 9.2 When n ≥ sr(R)+2, the natural map Stn(R) → Stn+1(R) is injective. In particular, when R is ﬁnite, the Steinberg group Stn(R) is ﬁnite for any n ≥ 3.

Proof. Let W (n, R) be the kernel of the natural map Stn(R) → Stn+1(R). When n ≥ sr(R)+2, the kernel W (n, R) is trivial (cf. Kolster [41], Theorem 3.1 and Cor. 2.10). When n is sufficient large, the Steinberg group Stn(R) is the universal central extension of En(R) (cf. [59], Proposition 5.5.1. page 240). Therefore, the kernel Stn(R) → En(R) is the second homology group H2(En(R); Z). When R is finite, both En(R) and H2(En(R); Z) are finite. Therefore, the group Stn(R) is finite for any n ≥ 3.

9.2 Rigidity of group homomorphisms on matrix groups Theorem 9.3 Suppose that G is a group satisfying that 1) G has a purely positive length function, i.e. there is a length function l : G → [0, ∞) such that l(g) > 0 for any infinite-order element g; and 2) any torsion abelian subgroup of G is finitely generated. Let R be an associative ring with identity and Stn(R) the Steinberg group. Suppose that S < Stn(R) is a finite-index subgroup. Then any group homomorphism f : Stn(R) → G has its image finite when n ≥ 3.

Proof. Since any ring R is a quotient of a free (non-commutative) ring ZhXi for some set X and Stn(R) is functorial with respect to the ring R, we assume without loss of generality that R = ZhXi. We prove the case S = Stn(R) ﬁrst. Let xij = hxij(r): r ∈ Ri, which is isomorphic to the abelian group R. Note that

[x12(1), x23(1)] = x13(1) and x13(1) commutes with x12(1) and x23(1). Lemma 5.2 implies any length function vanishes on x13(1). By Lemma 1.4, the length l(f(x13(1))) = 0. Note that xij(r) is conjugate to x13(r) for any r ∈ R and i, j satisfying 1 ≤ i =6 j ≤ n. Since l is purely postive, we get that f(x12(1)) is of ﬁnite order. Let I = ker f|x12 . Then I =6 ∅, as f(x12(1)) is of ﬁnite order. For any x ∈ I, and y ∈ R, we have

x12(xy) = [x13(x), x32(y)].

Therefore, f(x12(xy)) = [f(x13(x), f(x32(y)))] = 1 and thus xy ∈ I. Similarly, we have f(x12(yx)) = f([x13(y), x32(x)]) = 1. This proves that I is a (two-sided) ideal. Note that f(e12) = R/I is a torsion abelian group. By the

32 assumption 2), the quotient ring R/I is ﬁnite. Let Stn(R,I) be the normal subgroup of Stn(R) generated by xij(r),r ∈ I. There is a short exact sequence

1 → Stn(R,I) → Stn(R) → Stn(R/I) → 1.

Since R/I is finite, we know that Stn(R/I) is finite by Lemma 9.2. This proves that Im f is finite since f factors through Stn(R/I). For general finite-index subgroup S, we assume S is normal in Stn(R) after passing to a finite-index subgroup of S. A similar proof shows that S contains Stn(R,I) for some ideal I with the quotient ring R/I finite. Therefore, the image Im f is finite.

Theorem 9.4 Suppose that G is a group having a purely positive length function l. Let R be an associative ring of characteristic zero such that any nonzero ideal is of a finite index (eg. the ring of algebraic integers in a number field). Suppose that S < Stn(R) is a finite-index subgroup of the Steinberg group. Then any group homomorphism f : S → G has its image finite when n ≥ 3.

Proof. The proof is similar to that of Theorem 9.3. Let I = ker f|x12 , where x12 = S ∩hx12(r): r ∈ Ri. Since R is of characteristic zero and the length l(f(x12(k))) = 0 for ′ some integer k, we have f(x12(k)) is of finite order. Therefore, f(x12(k )) = 1 for some integer k′, which proves that the ideal I is nozero. Since I is of finite index in R, we get that Stn(R,I) is of finite index in S. This finishes the proof. Since the natural map Stn(R) → En(R) is surjective, any group homomorphism f : En(R) → G can be lifted to be a group homomorphism Stn(R) → G. Moreover, a finite- index subgroup E of En(R) is lifted to be a finite-index subgroup S of Stn(R). Theorem 0.4 and Theorem 0.6 follows Theorem 9.3 and Theorem 9.4, by inductive arugments on the subnormal series as those of the proofs of Theorem 0.3. Proof of Corollary 0.5 and Corollary 0.7. For Corollary 0.5, it is enough to check the two conditions for G in Theorem 0.4. Lemma 3.6 proves that G has a purely positive length function. When G is a CAT(0) group, (i.e. G acts properly and cocompactly on a CAT(0) space), then any solvable subgroup of G is finitely generated (and actually virtually abelian, see the Solvable Subgroup Theorem of [16], Theorem 7.8, page 249). When G is hyperbolic, it’s well-known that G contains finitely many conjugacy classes of finite subgroups and thus a torsion abelian subgroup is finite (see [16], Theorem 3.2, page 459). Birman-Lubotzky-McCarthy [7] proves that any abelian subgroup of the mapping class groups for orientable surfaces is finitely generated. Bestvina-Handel [6] proves that every solvable subgroup of Out(Fk) has a finite index subgroup that is finitely generated and free abelian. When G is the diffeomorphism group Diff(Σ,ω), there is a subnormal series (see the proof of Lemma 3.8)

1 ⊳ Ham(Σ,ω) ⊳ Diff0(Σ,ω) ⊳ Diff(Σ,ω), with subquoitents in Ham(Σ,ω), H1(Σ, R) and the mapping class group MCG(Σ). Any abelian subgroup of a finitely generated subgroup of these groups is finitely generated. Corrolary 0.7 follows Theorem 9.4 and Lemma 3.6. Remark 9.5 An infinite torsion abelian group may act properly on a simplicial tree (see [16], Example 7.11, page 250). Therefore, condition 2) in Theorem 0.4 does not hold for every group G acting properly on a CAT(0) (or a Gromov hyperbolic) space. We don’t know whether the condition 2) can be dropped.

33 10 Length functions on Cremona groups

Let k be a field and k(x1, x2, ··· , xn) be the field of rational functions in n indeterminates over k. It is well-known that the Cremona group Crn(k) is isomorphic to the automorphism group Autk(k(x1, x2, ··· , xn)) of the field k(x1, x2, ··· , xn).

Lemma 10.1 Let f : k(x1, x2, ··· , xn) → k(x1, x2, ··· , xn) be given by f(x1)= αx1, f(xi)= xi for some 0 =6 α ∈ k and any i =2, ··· , n. Then f lies in the center of a Heisenberg sub- −1 −1 group. In other words, there exists g, h ∈ Crn(k) such that [g, h]= ghg h = f, [g, f]= 1 and [h, f]=1.

Proof. Let g, h : k(x1, x2, ··· , xn) → k(x1, x2, ··· , xn) be given by

g(x1)= x1x2,g(xi)= xi(i =2, ··· , n) and −1 h(x1)= x1, h(x2)= α x2, h(xj)= xj(j =3, ··· , n). It can be directly checked that [g, h]= f, [g, f] = 1 and [h, f]=1.

n Lemma 10.2 Let l : Bir(Pk ) → [0, ∞) be a length function (n ≥ 2). Then l vanishes on n diagonal elements and unipotent elements of Aut(Pk ) = PGLn+1(k).

Proof. Let g = diag(a0, a1, ··· , an) ∈ PGLn+1(k) be a diagonal element. Note that l is subadditive on the diagonal subgroups. In order to prove l(g)=0, it is enough to prove that l(diag(1, ··· , 1, ai, 1, ··· , 1)) = 0, where diag(1, ··· , 1, ai, 1, ··· , 1) is the diagonal matrix with ai in the (i, i)-th position and all other diagonal entries are 1. But diag(1, ··· , 1, ai, 1, ··· , 1) is conjugate to diag(1, α, 1, ··· , 1) for α = ai. Lemma 10.1 implies that diag(1, α, 1, ··· , 1) lies in the center of a Heisenberg group. Therefore, l(diag(1, α, 1, ··· , 1)) = 0 by Lemma 5.2. This proves l(g)=0. The vanishing of l on unipotent elements follows Corollary 6.3 when the characteristic of k is not 2. When the characteristic of k is 2, any unipotent element A = I + u (where u is nilpotent) is of ﬁnite order. This means l(A) = 0.

Proof of Theorem 0.8. When k is algebraically closed, the Jordan normal form implies that any element g ∈ PGLn(k) is conjugate to the form sn with s diagonal and n the strictly upper triangular matrix. Moreover, sn = ns. Therefore, l(f) ≤ l(s)+ l(n). By Lemma 10.2, l(s)= l(n) = 0 and thus l(g) = 0.

2 Proof of Corollary 0.9. Let f : Bir(Pk) → G be a group homomorphism. Suppose that G has a purely positive length function l. By Theorem 0.8, the purely positive length function l on G will vanish on f(PGL3(k)). Since k is inﬁnite and PGL3(k) is a simple 2 group, we get that PGL3(k) lies in the ker f. By Noether-Castelnuovo Theorem, Bir(Pk) 2 is generated by PGL3(k) and an involution. Moreover, the Bir(Pk) is normally generated by PGL3(k). Therefore, the group homomorphism f is trivial. The general case is proved by an inductive argument on the subnormal series of a ﬁnite-index subgroup of G.

n Lemma 10.3 Let Bir(PR) (n ≥ 2) be the real Cremona group. Any length function l : n n Bir(Pk ) → [0, ∞), which is continuous on PSO(n+1) < Aut(PR), vanishes on PGLn+1(R).

34 Proof. By Lemma 10.2, the length function l vanishes on diagonal matrices of PGLn+1(R). Theorem 0.11 implies that l vanishes on the whole group PGLn+1(R).

Acknowledgements The author wants to thank many people for helpful discussions, including Wenyuan Yang on a discussion of hyperbolic groups, C. Weibel on a discussion on Steinberg groups of ﬁnite rings, Feng Su on a discussion of Lie groups, Ying Zhang on a discussion of translation lengths of hyperbolic spaces, Enhui Shi for a discussion on smooth measure- theoretic entropy.

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