Length functions on groups and rigidity Shengkui Ye January 25, 2021 Abstract 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(hgh−1)= l(g) for any h, g ∈ G; and (3) l(ab) ≤ l(a)+ l(b) for commuting elements a, b. Such length functions exist in many branches of mathematics, mainly as stable word lengths, stable norms, smooth measure-theoretic entropy, translation lengths on CAT(0) spaces and Gromov δ-hyperbolic spaces, stable norms of quasi-cocycles, rotation numbers of circle homeomorphisms, dynamical degrees of birational maps and so on. We study length functions on Lie groups, Gromov hyperbolic groups, arithmetic subgroups, matrix groups over rings and Cremona groups. As applica- tions, we prove that every group homomorphism from an arithmetic subgroup of a simple algebraic Q-group of Q-rank at least 2, or a finite-index subgroup of the elementary group En(R) (n ≥ 3) over an associative ring, or the Cremona group 2 Bir(PC) to any group G having a purely positive length function must have its image finite. Here G can be outer automorphism group Out(Fn) of free groups, mapping classes group MCG(Σg), CAT(0) groups or Gromov hyperbolic groups, or the group Diff(Σ,ω) of diffeomorphisms of a hyperbolic closed surface preserving an area form ω. 0.1 Introduction The rigidity phenomena have been studied for many years. The famous Margulis super- rigidity implies any group homomorphism between irreducible lattices in semisimple Lie groups of real rank rkR(G) ≥ 2 are virtually induced by group homomorphisms between the Lie groups. Therefore, group homomorphisms from ‘higher’-rank irreducible lattices arXiv:2101.08902v1 [math.GR] 22 Jan 2021 to ‘lower’-rank irreducible lattices normally have finite images. Farb, Kaimanovich and Masur [26] [39] prove that every homomorphism from an (irreducible) higher rank lattice into the mapping class group MCG(Σg) has a finite image. Bridson and Wade [17] showed that the same superrigidity remains true if the target is replaced with the outer automor- phism group Out(Fn) of the free group. Mimura [48] proves that every homomorphism from Chevalley group over commutative rings to MCG(Σg) or Out(Fn) has a finite image. Many other rigidity results can be found, e.g. [49] [18] [19] [33] [54] and [53]. In this article, we study rigidity phenomena with the notion of length functions. Let G be a group. We call a function l : G → [0, ∞) 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. 1 Such length functions exist in geometric group theory, dynamical system, algebra, algebraic geometry and many other branches of mathematics. For example, the following functions l are length functions (see Section 3 for more examples with details). • (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 dif- feomorphism 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 <C 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<G and a subnormal series 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<G and a subnormal series 1= Hn ⊳ Hn−1 ⊳ ··· ⊳ H0 = H satisfying that (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 homomor- phism 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).