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Home , Conjugacy class, Covering group, Group extension, Group homomorphism, Topological group

CLASSIFICATION OF THE CONJUGACY CLASSES OF SLf2(R)

CHRISTIAN TAFULA´

Abstract. In this paper, we classify the conjugacy classes of SLf2(R), the universal covering of PSL2(R). For any non-central element α ∈ SLf2(R), we show that its conjugacy class may be determined by three invariants: (i) Trace: the trace (valued in the of R+) of its α in PSL2(R); (ii) Direction type: the sign behavior of the induced self- of R 1 determined by the lifting SLf2(R) y R of the action PSL2(R) y S ; (iii) Length-sharp: a conjugacy invariant version of the length in S. Zhang [12], introduced by S. Mochizuki [9].

Contents 0. Introduction...... 1 1. CP-extensions of groups ...... 3 1 2. The group Homeo+(S )...... 14 g 1 3. Invariants in Homeog +(S )...... 21 4. Conjugacy classes of SLf2(R)...... 25 References...... 31

0. Introduction Our goal in this paper is to classify the conjugacy classes of the universal of PSL2(R), namely, SLf2(R). Based on the group-theoretic proof of Szpiro’s theorem as described by Zhang [12] (originally due to Bogomolov et al. [1]), we start 1 by identifying PSL2(R) with a of Homeo+(S ), the group of orientation- preserving self- of S1. This is done via the faithful continuous action 1 induced by PSL2(R) on S as described in Section 4. Then, via the natural quotient map R 3 x 7→ x ∈ S1 ' R/Z, we show that the topological of 1 1 Homeo+(S ) is π1(Homeo+(S )) ' Z and then describe its universal covering group 1 Homeo] +(S ) as a subgroup of Homeo+(R), the group of self-homeomorphisms of R which are increasing. The

1 1 1 Z Homeo] +(S ) Homeo+(S ) 1 is shown to be a CP-extension, a notion which is defined in the discussion following Definition 1.1. The notion of a CP-extension is characterized in Theorem 1.12 and

2010 Mathematics Subject Classification. Primary 20E45; Secondary 22E46. Key words and phrases. SL(2, R), conjugacy classes, group extensions.

1 2 CHRISTIAN TAFULA´

discussed at length in Section 1. Denote the image of Z in the above by Z. 1 In Lemma 2.5, we show that this subgroup is actually the of Homeo] +(S ). A particular consequence of Theorem 1.12, which allows us to calculate the conjugacy 1 classes of SLf2(R) explicitly, is that the set of conjugacy classes of Homeo] +(S ) admits 1 a natural structure of Z-torsor over the set of conjugacy classes of Homeo+(S ). This 1 property is inherited by arbitrary of Homeo] +(S ) that contain the subgroup 1 Z. More explicitly, we show that the conjugacy class of an element α ∈ Homeo] +(S ) is determined uniquely by the following data: • an element ζ ∈ Z, • the canonical lifting (as defined in Proposition 2.10) of a certain representative of 1 the conjugacy class of its image α ∈ Homeo+(S ). This is stated in Corollary 2.11. The two main concepts used in the proof are the notions of CP- of groups (Definition 1.1) and quasi-homomorphisms (Definition 1.14 — cf. Fujiwara [5]). Corollary 2.11 is derived from the somewhat more technical Theorem 1.21, which, roughly speaking, states that “central group extensions whose may be separated from the identity by quasi-homomorphisms are CP-extensions”. Separation is defined in Subsection 1.5. Next, we classify the non-central conjugacy classes of SLf2(R) completely by considering the trace (which is motivated by the usual classification of the conjugacy classes of SL2(R) as elliptic, parabolic, or hyperbolic — cf., for instance, K. Conrad [3]), together with two further invariants of conjugacy classes in SLf2(R), which 1 arise from the structure of SLf2(R) as a subgroup of Homeo] +(S ). The invariants considered are the following: a b (i) Trace: tr(α) := |a + d|, where [ c d ] = α ∈ PSL2(R) is the image of α ∈ SLf2(R) via the natural quotient map;

1 (ii) Direction type: Elements α ∈ Homeo] +(S ) will be constructed in such a way that the function R 3 x 7→ α(x) − x ∈ R is invariant with respect to the translation R 3 x 7→ x + 1 ∈ R. The sign behavior of α(x) − x can be classified as follows: strictly positive (the forward direction type), strictly negative (the backward direction type), positive but not strictly positive (the semi-forward direction type), negative but not strictly negative (the semi-backward direction type), or alternating in sign (the alternating direction type). (iii) Length-sharp: As was mentioned above, α(x) − x is invariant under translation by 1; hence, in to determine the behavior of α, it suffices to restrict our

attention to x ∈ [0, 1], which is compact. In particular, `(α) := supx∈[0,1] |α(x)− x| is a well-defined . We then define the length-sharp function by taking `](α) := {b`(α)c, d`(α)e} ⊆ N. The direction type (cf. (ii)) and length-sharp function (cf. (iii)) are invariant under 1 conjugation in Homeo] +(S ) (Lemma 3.2 and Proposition 3.4, respectively). The length-sharp function, which we describe in Subsection 3.2, is based on the exposition of Bogomolov’s proof of Szpiro’s theorem by Mochizuki [9]. A table at the end of this paper shows how (i), (ii), and (iii) determine the conjugacy classes of non-central elements of SLf2(R). CLASSIFICATION OF THE CONJUGACY CLASSES OF SLf 2(R) 3

0.1. Notation. Write N for the set of non-negative . For x ∈ R, we use the standard notations bxc := max{k ∈ Z | k ≤ x} and dxe := min{k ∈ Z | k ≥ x} for the floor and ceiling functions, respectively. 0.1.1. Groups. For a group G, we denote its by e ∈ G and the center of G by Z(G). For a S ⊆ G, we denote the commutant (or centralizer) of S in G by ZG(S) := {g ∈ G | gs = sg, ∀s ∈ S}. Given g1, ∈ G, we say that g1 −1 is conjugate to g2 if there exists h ∈ G such that g1 = hg2h , in which case we write g1 ∼ g2. Conjugacy is an ; we denote the conjugacy class of g ∈ G by [g] := {h ∈ G | g ∼ h} and the set of all conjugacy classes of G by C`(G). For any two elements g, h ∈ G, we denote their by [g, h] := ghg−1h−1. For elements gi ∈ G, indexed by some set I 3 i, the subgroup generated by the gi’s will be denoted by hgi | i ∈ Ii. For N E G a , we denote the associated natural quotient map by G 3 g 7→ g ∈ G/N when no ambiguity is present.

a b 0.1.2. Special . An element ( c d ) ∈ SL2(R) is a with real entries satisfying ad − bc = 1, and we denote its image in PSL2(R) := SL2(R)/{±Id}, where 1 0 a b  −a −b  Id := ( 0 1 ), by [ c d ] = −c −d ∈ PSL2(R). Since the trace function on SL2(R) is a b given by tr(( c d )) = a + d, the function M 7→ |tr(M)| naturally factors through a b a b PSL2(R); thus, we define the trace function on PSL2(R) by tr([ c d ]) := tr(( c d )) . 0.1.3. Groups of self-homeomorphisms. For a X, write Homeo(X) for the group of self-homeomorphisms of X and Homeo0(X) ⊆ Homeo(X) for the subgroup of self-homeomorphisms that are isotopic to the identity. In general, given two topological spaces X,Y , the compact-open topology on C(X,Y ), the set of continuous maps from X to Y , is defined to be the topology generated by sets of the form V (K,U) := {f ∈ C(X,Y ) | f(K) ⊆ U}, where K ⊆ X is compact and U ⊆ Y is open. We regard Homeo(X) ⊆ C(X,X) as being equipped with the topology induced by the compact-open topology on C(X,X). Thus, Homeo0(X) is the connected of Homeo(X) that contains the identity. In the particular 1 1 1 case of the circle S := R/Z and the real line R, we write Homeo+(S ) := Homeo0(S ) and Homeo+(R) := Homeo0(R). 0.1.4. Torsors. Let C be a that admits finite products. For an object Y and a G in C, we shall say Y is a (left) G-torsor if there is a (left) action G y Y , written σ : G × Y → Y , such that the

σ × pG×Y/Y : G × Y −→ Y × Y, where pG×Y/Y : G × Y → Y denotes the projection to the second factor, is an . This is a natural generalization of the notion of principal G-bundles for smooth .

1. CP-extensions of groups 1.1. CP-homomorphisms. We start with a definition. Definition 1.1 (CP-). A homomorphism of groups φ : G → H is called commutant-preserving homomorphism (or a CP-homomorphism, for short) if ∀x, y ∈ G, φ(x)φ(y) = φ(y)φ(x) =⇒ xy = yx. 4 CHRISTIAN TAFULA´

Remark 1.2. For any φ : G → H, it holds that ZG(x) ⊆ −1  φ ZH (φ(x)) for all x ∈ G; thus, φ is a CP-homomorphism exactly when the −1  T converse holds as well, i.e., ZG(x) = φ ZH (φ(x)) . Since ZG(S) = x∈S ZG(x) for any subset S ⊆ G, a CP-homomorphism φ : G → H satisfies −1  (1.1) ∀S ⊆ G, ZG(S) = φ ZH (φ(S)) ,

hence its name. In particular, ZG(ker(φ)) = G, i.e., ker(φ) ⊆ Z(G). We briefly recall the following basic facts on group extensions: Given two groups Q, N, a group extension G of Q by N consists of an injective homomorphism from N to G and a surjective homomorphism φ from G to Q such that the resulting sequence φ (1.2) 1 N G Q 1 is exact. If we identify N with its image in G, then N = ker(φ). An extension is called central if N ⊆ Z(G). Definition 1.3 (CP-extensions and CP-groups). We shall say that an extension 1 (1.2) is a CP-extension if φ : G  Q is a CP-homomorphism. (Thus, by Remark 1.2, every CP-extension is central.) A group G is called a CP-group if the quotient map G  G/Z(G) is a CP-extension. Several characterizations of CP-extensions are given in Theorem 1.12. Notice that CP-extensions are necessarily central. The first relevant properties of CP- homomorphisms are the following: Lemma 1.4 (Basic properties of CP-homomorphisms). Let φ : G → H and ψ : H → K be group homomorphisms. Then the following hold: (i) If φ and ψ are CP-homomorphisms, then the composite ψ ◦ φ : G → K is also a CP-homomorphism. (ii) If the composite ψ ◦ φ is a CP-homomorphism, then φ is a CP-homomorphism. Moreover, if φ is surjective, then ψ is also a CP-homomorphism. (iii) If φ is CP-homomorphism, then for any group homomorphism ξ : A → H, the map A ×H G → A induced by φ is a CP-homomorphism. (iv) If G is a CP-group, then any central extension 1 → N → G → Q → 1 is a CP-extension. Proof. We prove the items separately, starting from item (i). The condition that φ be a CP-homomorphism is equivalent to the condition that φ([x, y]) = eH =⇒ [x, y] = eG, for any commutator [x, y] ∈ G, where eG, eH , eK denote the identity elements of the respective groups. Thus, since both φ and ψ are CP-homomorphisms and φ([x, y]) = [φ(x), φ(y)],

ψ(φ([x, y])) = eK =⇒ φ([x, y]) = eH =⇒ [x, y] = eG, so ψ ◦ φ is a CP-homomorphism. Next, we consider item (ii). Suppose that ψ ◦ φ is a CP-homomorphism. Then, for any x, y ∈ G, we have

φ([x, y]) = eH =⇒ ψ(φ([x, y])) = eK =⇒ [x, y] = eG,

1In other words, a CP-extension amounts to a surjective CP-homomorphism. CLASSIFICATION OF THE CONJUGACY CLASSES OF SLf 2(R) 5

so φ is a CP-homomorphism. If, in , φ is surjective, then for any a, b ∈ H there exist x, y ∈ G such that φ(x) = a and φ(y) = b. Thus,

ψ([a, b]) = eK =⇒ ψ(φ([x, y])) = eK =⇒ [x, y] = eG, so ψ is a CP-homomorphism. Next, we consider item (iii). Consider the commutative diagram

A ×H G G

p φ ξ A H.

Since A ×H G = {(a, g) ∈ A × G | ξ(a) = φ(g)}, given (a1, g1), (a2, g2) ∈ A ×H G, we have

a1a2 = a2a1

=⇒ ξ(a1)ξ(a2) = ξ(a2)ξ(a1)

=⇒ φ(g1)φ(g2) = φ(g2)φ(g1)

=⇒ g1g2 = g2g1 (because φ is a CP-homomorphism),

therefore a1a2 = a2a1 implies (a1, g1)(a2, g2) = (a2, g2)(a1, g1). Finally, item (iv) follows from item (ii), by considering the composite of surjections G  G/N  G/Z(G).  1.2. Induced map on conjugacy classes. Consider the (1.2). We denote conjugacy equivalence in G by ∼G, and G-conjugacy classes by a subscript G. Thus, for g ∈ G, we have [g]G = {a ∈ G | a ∼G g}. We use similar notational conventions for Q. The surjection φ induces a surjective map of sets Φ : C`(G) 3 −1 [g]G 7→ [φ(g)]Q ∈ C`(Q). Our goal (Proposition 1.8) is to describe Φ ([q]Q) for each conjugacy class [q]Q ∈ C`(Q). We start with the general case, where N is an arbitrary normal subgroup of G, and later we impose the hypothesis that N ⊆ Z(G). −1 For each q ∈ Q, for the rest of this subsection, choose and fix a lifting qe ∈ φ (q) ⊆ G. Thus, every element in G may be written in the form kqe, for a uniquely determined (k, q) ∈ N × Q. Write I := C`(Q). Choose and fix a representative element qi ∈ Q for each i ∈ I, so C`(Q) = {[qi]Q | i ∈ I}.

Lemma 1.5. For every g ∈ G, there exists a unique i ∈ I such that g ∼G kqei, for some (not necessarily unique) k ∈ N.

Proof. Write g = k1qe. By definition, there is a unique i ∈ I such that q ∼Q qi, i.e., −1 −1 −1 ∃p ∈ Q such that q = pqip . Since φ(pq^ip ) = φ(peqei pe ), there exists k2 ∈ N such −1 −1 −1 that pq^ip = k2 · peqei pe . Furthermore, since N is normal, k3 := pe k1k2 pe ∈ N. −1 −1 −1 Hence, g = k1qe = k1 · pq^ip = k1k2 · peqei pe = pe(k3qei) pe , so g ∼G k3qei, as desired.  Therefore, in order to determine the fibers of Φ, it suffices to analyse, for each i ∈ I, the conditions under which kqei ∼ jqei for k, j ∈ N. To this end, we consider the following. 6 CHRISTIAN TAFULA´

Definition 1.6 (Twisted conjugation). Let N E G be a fixed normal subgroup of a group G. For each g ∈ G, we define g-twisted conjugation in N to be the following equivalence relation: for k, j ∈ N, g −1 (1.3) k ≈g j ⇐⇒ ∃a ∈ G | k = aj(a ) , g −1 where we write a := gag . When k ≈g j, we say that k and j are g-twisted conjugate in N, and, when there is no ambiguity with respect to N, we write [[k]]g ∈ N/≈g for the set of elements of N that are g-twisted conjugate to k. Here, we note that this relation is well-defined, for if k, j, ` ∈ N are such that k = aj(ag)−1, j = b`(bg)−1 for some a, b ∈ G, then k = ek(eg)−1, j = a−1k((a−1)g)−1, k = (ab)`(ab)g−1,

so ≈g is reflexive, symmetric and transitive. The following are basic properties of twisted conjugation. Lemma 1.7. Let g ∈ G. Then, for any k, j ∈ N and p ∈ G, the following hold:

(i) kg ∼G jg ⇐⇒ k ≈g j; (ii) k ≈pg j ⇐⇒ kp ≈g jp; −1 −1 (iii) k ≈pgp−1 j ⇐⇒ p kp ≈g p jp; Proof. For item (i), we have −1 kg ∼G jg ⇐⇒ ∃a ∈ G | kg = ajga ⇐⇒ ∃a ∈ G | k = ajga−1g−1 = aj(ag)−1

⇐⇒ k ≈g j. Item (ii) follows immediately from item (i):

k ≈pg j ⇐⇒ kpg ∼G jpg ⇐⇒ kp ≈g jp. −1 Item (iii) follows immediately by conjugating by p .  −1 We can now apply Lemmas 1.5 and 1.7 (i) to describe the fibers Φ ([qi]Q) of the map Φ : C`(G) 3 [g]G 7→ [φ(g)]Q ∈ C`(Q), as follows: For each i ∈ I, write Ψ : N 3 k 7→ [[k]] ∈ Λ := N/ ≈ . Then, for each λ(i) ∈ Λ , choose and fix a i qei i qei i −1 (i) representative element κλ(i) ∈ Ψi (λ ). Thus, we have: Proposition 1.8. It holds that  (i) C`(G) = [κλ(i) qei]G | i ∈ I, λ ∈ Λi . In particular, if the q -twisted conjugacy relations are trivial (i.e., N/≈ = N for ei qei every i ∈ I), then the map N × C`(Q) 3 (k, [q ] ) 7→ κ q  ∈ C`(G) is bijective. i Q Ψi(k) ei G Remark 1.9. One way to interpret Proposition 1.8 is by viewing the qei-twisted conjugacy relations as encoding the obstructions preventing the map N × C`(Q) 3 (k, [q ] ) 7→ κ q  ∈ C`(G) from being injective. i Q Ψi(k) ei G CLASSIFICATION OF THE CONJUGACY CLASSES OF SLf 2(R) 7

The problem with the description of C`(G) given in Proposition 1.8 lies in the −1 fact that it depends on both the choice of liftings qe ∈ φ (q) and the choice of representatives for [q]Q in Q. The properties of Lemma 1.7 (ii), (iii) describe how −1 different liftings (say q,e qb ∈ φ (q)) and different representatives for conjugacy classes in Q (say r, s ∈ [q]Q) may give rise to different equivalence relations on N (i.e., N/≈ ⇐⇒/N/≈ ). The situation is substantially ameliorated when N is central. re sb 1.3. Properties of CP-extensions. We continue to consider a group extension

φ 1 N G Q 1. If N is a central subgroup of G, then the multiplication by an element in N to the left induces an action (1.4) N × C`(G) 3 (k, [g]) 7→ [kg] ∈ C`(G). On the other hand, for k ∈ G, in order for the multiplication by k to the left to be well defined as a map of conjugacy classes, it is necessary and sufficient that −1 (1.5) ∀g ∈ G, ∀p ∈ G, k(pgp ) ∼G kg, which is clearly the case when N is central. Moreover, the converse holds. Lemma 1.10. An element k ∈ G satisfies (1.5) if and only if k ∈ Z(G). Proof. If k ∈ Z(G), then (1.5) clearly holds. For the other direction, suppose (1.5) holds for k. Since this condition applies to all g, p ∈ G, it applies, in particular, −1 for g = k , which implies [k, p] ∼G e for all p ∈ G. Since [e]G = {e}, that means [k, p] = e for every p ∈ G, therefore k ∈ Z(G).  If N is central, then, using the above action, we may describe the fibers of the map Φ : C`(G)  C`(Q) without referencing representatives or liftings. Indeed, this is a consequence of the next lemma.

Lemma 1.11. Suppose N ⊆ Z(G), and define Sg := {[a, g] | a ∈ G} ∩ N for each g ∈ G. Then, the following hold:

(i) For each g ∈ G, the set Sg is a subgroup of N, and N/≈g = N/Sg. (ii) We have Sg = Skg for every k ∈ N and g ∈ G. (iii) We have Sg = Spgp−1 for every p, g ∈ G. (iv) For g, h ∈ G, if φ(g) ∼Q φ(h) then Sg = Sh. (v) For a conjugacy class [q] ∈ C`(Q), write S := S , for some lifting q ∈ φ−1(q). Q [q] qe e Then, the action N y C`(G) described in (1.4) induces a free, transitive action −1 N/S[q] y Φ ([q]Q). Proof. For item (i), since N ⊆ Z(G), we have [a, g][b, g] = a[b, g]a−1 [a, g] = [ab, g], [a, g]−1 = a−1[a, g]−1a = [a−1, g], thus, Sg is a group. From the definition of ≈g in (1.3) and the assumption that −1 N ⊆ Z(G), we have k ≈g j ⇐⇒ kj ∈ Sg for k, j ∈ N, and thus N/≈g = N/Sg. Item (ii) (resp. (iii)) follows from Lemma 1.7 (ii) (resp. (iii)). 8 CHRISTIAN TAFULA´

For item (iv), the condition φ(g) ∼Q φ(h) implies that there are k ∈ N, p ∈ G such that g = k php−1. Then, item (iv) follows from the items (ii) and (iii). Finally, for item (v), notice that for each conjugacy class [q]Q ∈ C`(Q), we deduce from item (iv) that the group S depends neither on the choice of liftings q ∈ φ−1(q) qe e nor on the choice of a representative for [q]Q; that is, S[q] is well-defined. From item (i), we have N/≈ = N/S , therefore, by Lemma 1.7 (i), qe [q] StabN ([qe]G) : = {k ∈ N | kqe ∼G qe} = {k ∈ N | k ≈ e} = S , qe [q] −1 implying that the induced action N/S[q] y Φ ([q]Q) is well-defined and free. More- over, in the notation from the discussion leading up to Proposition 1.8, letting qi ∈ Q, i ∈ I be such that q ∼Q qi, Proposition 1.8 says that N/S × {[q ] } 3 k, [q ]  7−→ κ q  ∈ Φ−1([q] ) ⊆ C`(G) [q] i Q i Q Ψi(k) ei G Q is a one-to-one correspondence, amounting to the fact that the action is transitive.  With Lemma 1.11 (v), we obtain a complete description of C`(G) in terms of N and C`(Q), depending only on φ, which is necessary for determining S[q]. Categorically, −1 we say that Φ ([q]Q) is a N/S[q]-torsor in Set. When G is a CP-extension of Q, it turns out that S[q] is trivial for every [q]Q ∈ C`(Q), as will be shown in the next theorem. Thus, in this situation, N acts freely on C`(G), and the orbits of this action −1 are exactly the Φ ([q]Q) for [q]Q ∈ C`(Q); i.e., C`(G) is an N-torsor over C`(Q) (along Φ), meaning that it is a (N × C`(Q))-torsor in the slice category Set/C`(Q).

φ Theorem 1.12 (Characterizations of CP-extensions). Let 1 → N → G → Q → 1 be an exact sequence of groups. Then, the following are equivalent: (i) G is a CP-extension of Q by N. −1 −1 (ii) For any subset T ⊆ Q, it holds ZG(φ (T )) = φ (ZQ(T )). (iii) N ∩ {[a, b] | a, b ∈ G} = {e}. (iv) The twisted conjugacy classes in N are trivial; i.e., ∀g ∈ G, N/≈g = N. (v) C`(G) is an N-torsor over C`(Q) (along Φ). (vi) For p, q ∈ Q, it holds

pq = qp (in Q) =⇒ peqe = qepe (in G), −1 −1 for any liftings pe ∈ φ (p), qe ∈ φ (q). In other words, any liftings of elements that commute with each other also commute with each other. Proof. The equivalences (i) ⇐⇒ (iii) ⇐⇒ (vi) follow by definition, and (i) ⇐⇒ (ii) follows from (1.1) together with the fact that φ : G  Q is surjective. We prove the remaining implications separately. • (iii) ⇐⇒ (iv): Since N is a normal subgroup of G, we have [g, k] ∈ N ∩ {[x, y] | x, y ∈ G} for any g ∈ G and k ∈ N. Then, the condition (iii) implies that [g, k] = e for any g ∈ G and k ∈ N, hence N ⊆ Z(G). For arbitrary g ∈ G, let k, j ∈ N such that k ≈g j. Then, from the definition of ≈g in (1.3), there is a ∈ G such that k = aj(ag)−1. Since N ⊆ Z(G), we have kj−1 = [a, g] ∈ N ∩ {[x, y] | x, y ∈ G} = {e}, hence k = j. We proved that (iii) =⇒ (iv). CLASSIFICATION OF THE CONJUGACY CLASSES OF SLf 2(R) 9

For the converse, assume (iii) is false. Then, there is a, b ∈ G such that [a, b] = k ∈ N, with k 6= e. This is the same as aba−1 = kb, which, by Lemma 1.7 (i), implies k ≈b e, therefore N/≈b 6= N. Hence, (iv) =⇒ (iii). • (iv) ⇐⇒ (v): The implication (iv) =⇒ (v) follows from Lemma 1.11 (v). For the converse, notice that in order for the action N y C`(G) to be well-defined, by −1 Lemma 1.10, we must have N ⊆ Z(G). Moreover, since N acts freely on Φ ([q]Q) for every [q]Q ∈ C`(Q), we have, for every g ∈ G and k 6= j ∈ N, that [kg]G 6= [jg]G, i.e., kg 6∼G jg. By Lemma 1.7 (i), this implies that for every `, m ∈ N, it holds ` ≈g m if and only if ` = m, which means N/≈g = N. We proved that (v) =⇒ (iv).  Remark 1.13 (Examples of CP-extensions). Group extensions with the properties described in Theorem 1.12 are plenty in the literature, generally appearing in the form of item (vi). We give three classes of groups as examples. n n (i)H omeo] 0(T ) : For each n ≥ 1, write Homeo0(T ) for the of self-homeomorphisms of the n-dimensional which are isotopic to the identity, equipped with the compact-open topology. This group admits a uni- n n n n versal covering Homeo] 0(T ), which, by writing T = R /Z , may be identified with those self-homeomorphisms of Rn which commute with translation by an el- n n n n ement of the Z ⊆ R . The group extension Homeo] 0(T )  Homeo0(T ) is a CP-extension (cf. Remarque 2.12, p. 22 of Herman [8]). In Section 2, we will study in detail the particular case for n = 1, however, we follow different methods, closer to what is presented in Ghys [7].

OI  (ii) Spf2n(R) : Following Chapter 2 of de Gosson [4], for n ≥ 1, write Ω := −IO for the standard symplectic matrix, where O and I are the n × n zero and identity matrices, respectively. A 2n × 2n real matrix A is called symplectic if ATΩA = AΩAT = Ω, and we call the set of all 2n × 2n symplectic matrices the real Sp2n(R). This is a real, non-compact, connected , with topological fundamental group isomorphic to Z. Following Barge–Ghys [2], its universal covering Spf2n(R) may be realized through liftings of self-homeomorphisms induced by the natural action of Sp2n(R) on the Lagrangian Grassmannian Λn ' U(n)/O(n) (cf. Propositions 43 and 46, Section 2 of de Gosson [4]). As with the previous case, Spf2n(R)  Sp2n(R) is a CP-extension (cf. item (i) of Proposition 2.3 from Barge–Ghys [2]). In Section 4, we will study the particular case n = 1, where we have Sp2(R) = SL2(R), using Theorem 1.12 (v) to calculate the conjugacy classes of SLf2(R) explicitly. (iii) Z(G) torsion-free, G/Z(G) torsion : For our final example, suppose G is group such that Z(G) is torsion-free, meaning that if k ∈ Z(G), then kn 6= e for all non-zero n ∈ Z. Moreover, suppose G/Z(G) is a torsion group, meaning that every element g ∈ G/Z(G) has finite order, so that there is n = n(g) ∈ N≥1 such that gn is the identity in G/Z(G). Then, G is a CP-group, or, equivalently, G  G/Z(G) is a CP-extension. To see how this is the case, we use Theorem 1.12 “(i) ⇐⇒ (iii)”. Take k ∈ Z(G) ∩ {[g, h] | g, h ∈ G}, so that k = [a, b] for some a, b ∈ G. Then, we have aba−1 = kb, and since k is central, it holds 10 CHRISTIAN TAFULA´

abna−1 = knbn for every n ∈ Z. However, that means kn = [a, bn]. Since n b ∈ G/Z(G) has finite order, there must be some n ∈ N≥1 for which b ∈ Z(G), implying that kn = e. Since Z(G) is torsion-free, this only happens if k = e, implying that Z(G) ∩ {[g, h] | g, h ∈ G} = {e}, and hence, by Theorem 1.12, G  G/Z(G) is a CP-extension. The converse of this statement is not true (cf. Remark 2.9), but this is the prototypical example to be kept in mind as we go on to the rest of this section. The notion of quasi-homomorphisms (cf. Definition 1.14) will be introduced with the intent of loosening up in a precise sense the condition of G/Z(G) being a torsion group (e.g., through the role played by a non-trivial defect). See Lemma 1.18 together with Theorem 1.21. 1.4. Vector-valued quasi-homomorphisms. Quasi-homomorphisms are gener- ally defined as real-valued functions from a group G that fail to be a homomorphism to R by, at most, a bounded amount (cf. Barge–Ghys [2], Fujiwara [5], Gambaudo– Ghys [6], Ghys [7]). We consider a slightly more general version of this definition, allowing arbitrary Banach spaces (V, k·k) (i.e., archimedean normed vector spaces over R or C which are complete with respect to their norm) instead of just R. Definition 1.14. Let G be a group and (V, k·k) a . A map ϕ : G → V is called a quasi-homomorphism if

Dϕ := sup kϕ(gh) − ϕ(g) − ϕ(h)k < +∞. g,h∈G

k The number Dϕ ∈ R≥0 is called the defect of ϕ. If, in addition, it holds ϕ(g ) = kϕ(g) for every k ∈ Z, then ϕ is said to be homogeneous. The first important property of such maps is that one can always “homogenize” a quasi-homomorphism. That is:

Lemma 1.15 (Homogenization). Let ϕ : G → V be a quasi-homomorphism. For each g ∈ G, write ϕ(gn) (1.6) ϕ(g) := lim . b n→+∞ n Then, ϕb : G → V is a well-defined homogeneous quasi-homomorphism, and we call it the homogenization of ϕ.

Proof. Let Dϕ denote the defect of ϕ. Fixing g ∈ G, for every integer n ≥ 2 we have n n−1 kϕ(g ) − ϕ(g ) − ϕ(g)k ≤ Dϕ. Thus, since n n n−1  n−1  kϕ(g ) − nϕ(g)k = ϕ(g ) − ϕ(g ) − ϕ(g) + ϕ(g ) − (n − 1)ϕ(g) n−1 ≤ Dϕ + ϕ(g ) − (n − 1)ϕ(g) , n one can use induction to show that kφ(g ) − nϕ(g)k < (n − 1)Dϕ for every n ≥ 2. Since V is Banach, hence complete, to show that (1.6) converges, it suffices to n show that (ϕ(g )/n)n≥1 is a . For that aim, let ε > 0 and set N := d2Dϕ/εe. Taking n, m ≥ N, we have n m n m ϕ(g ) ϕ(g ) kmϕ(g ) − nϕ(g )k − = n m nm CLASSIFICATION OF THE CONJUGACY CLASSES OF SLf 2(R) 11

n nm  mn m  mϕ(g ) − ϕ(g ) + ϕ(g ) − nϕ(g ) = nm (m − 1) + (n − 1) ≤ D nm ϕ  1 1  ≤ + D ≤ ε. n m ϕ Therefore, the limit at (1.6) is well-defined. To see that ϕb is a quasi-homomorphism, take g, h ∈ G. For n ≥ 2, we have n n n n n ϕ((gh) ) ϕ(g ) ϕ(h ) kϕ((gh) ) − nϕ(gh)k kϕ(g ) − nϕ(g)k − − ≤ + + n n n n n kϕ(hn) − nϕ(h)k + + kϕ(gh) − ϕ(g) − ϕ(h)k n ≤ 4Dϕ, n n n thus, as ϕ((gh) ) − ϕ(g ) − ϕ(h ) → ϕ(gh) − ϕ(g) − ϕ(h) as n → +∞, we conclude that n n n b b b ϕb has defect at most 4Dϕ, and hence it is a quasi-homomorphism. For homogeneity, start by noting that, for every g ∈ G and m ≥ 1, we have ϕ(gmn) ϕ(gmn) ϕ(gm) = lim = m lim = m ϕ(g). b n→+∞ n n→+∞ mn b

Since ϕb(e) = limn→+∞ ϕ(e)/n = 0V ∈ V, where 0V is the origin point of V, for every n ≥ 1 it holds D ≥ ϕ(gng−n) − ϕ(gn) − ϕ(g−n) = n ϕ(g) + ϕ(g−1) ϕb b b b b b −1 k implying that ϕb(g ) = −ϕb(g). Consequently, we deduce that ϕb(g ) = k ϕb(g) for every k ∈ Z, showing that ϕb is homogeneous.  Lemma 1.16. For quasi-homomorphisms ϕ, ψ : G → V, it holds ϕb = ψb if and only if supg∈G kϕ(g) − ψ(g)k < +∞.

Proof. If supg∈G kϕ(g) − ψ(g)k < +∞, then, for every g ∈ G, it holds 1 lim kϕ(gn) − ψ(gn)k = 0, n→+∞ n from where it is clear that ψb = ϕb. For the converse, we have, for every n ≥ 1, n n ϕ(g ) ψ(g ) 1 n n kϕ(g) − ψ(g)k ≤ ϕ(g) − + − ψ(g) + kϕ(g ) − ψ(g )k . n n n n As shown in the proof of Lemma 1.15, we have kϕ(g ) − nϕ(g)k ≤ (n−1)Dϕ for every g ∈ G and n ≥ 2, where Dϕ denotes the defect of ϕ. Hence, letting n → +∞ in the above inequality, we deduce that, for every g ∈ G, it holds kϕ(g) − ψ(g)k ≤ Dϕ +Dψ, which does not depend on g. Therefore, our claim follows.  Following Fujiwara [5], for a group G and a fixed Banach space V, the set of all quasi-homomorphisms from G to V forms a vector space (over R or C, depending on V), which we denote by QH(G, V). Writing BDD(G, V) for the space of bounded 2 functions, it is clear that BDD(G, V) ⊆ QH(G, V). Then, writing HQH(G, V) for 2 i.e., maps ϕ : G → V for which there is C = C(ϕ) ∈ R such that kϕ(g)k < C for all g ∈ G. 12 CHRISTIAN TAFULA´ the space of homogeneous quasi-homomorphisms, Lemmas 1.15 and 1.16 may be summarized in the isomorphism QH(G, V)/BDD(G, V) ' HQH(G, V), which is given by the homogenization construction. Note that our choice of notation slightly differs from that of Fujiwara [5]. Two properties of homogeneous quasi- homomorphisms that will be relevant to us are the following: Lemma 1.17. Homogeneous quasi-homomorphisms are class functions, i.e., invari- ant under conjugation.

Proof. Let ϕ : G → V be a homogeneous quasi-homomorphism with defect Dϕ. For any g, h ∈ G and n ≥ 1, we have, from homogeneity n −1 n ϕ(gh g ) − ϕ(h ) n −1 n −1 = ϕ(gh g ) − ϕ(g) − ϕ(h ) − ϕ(g ) n −1 n −1 n n ≤ ϕ(gh g ) − ϕ(gh ) − ϕ(g ) + kϕ(gh ) − ϕ(g) − ϕ(h )k

≤ 2Dϕ, which is independent of n. Thus, since n −1 n −1 ϕ(gh g ) − ϕ(h ) = n ϕ(ghg ) − ϕ(h) , −1 we conclude that ϕ(h) = ϕ(ghg ).  By considering an exact sequence 1 → N → G → Q → 1 with N := Z(G), the next lemma may be viewed in comparison with Theorem 1.12 (iii).

Lemma 1.18. For any homogeneous quasi-homomorphism ϕ : G → V, we have Z(G) ∩ {[g, h] | g, h ∈ G} ⊆ ker(ϕ). Proof. Let g, h ∈ G and suppose that [g, h] = k ∈ Z(G). That means ghg−1 = kh, implying that, for every n ∈ Z, it holds ghng−1 = knhn, and thus [g, hn] = kn. Since n ϕ is homogeneous, we have ϕ([g, h ]) = n ϕ(k). However, letting Dϕ be the defect of ϕ, we have, from homogeneity and Lemma 1.17, n −1 −n n −1 −n n −1 −n Dϕ ≥ ϕ(gh g h ) − ϕ(g) − ϕ(h g h ) = ϕ(gh g h ) , and thus n kϕ(k)k ≤ Dϕ for all n ∈ Z, which implies ϕ(k) = 0V.  The relation to Remark 1.13 (iii) comes from the fact that, if g ∈ G has finite order, then g necessarily lies in the kernel of any homogeneous quasi-homomorphism from G. The converse, however, is not necessarily true, and thus we cannot deduce Remark 1.13 (iii) from Lemma 1.18. To be able to conclude a similar statement, we will need the notion of quasi-separation, which we introduce now. 1.5. Quasi-separation and CP-groups. Following Fujiwara (cf. Section 6 of [5]) and Polterovich–Rudnick [10], let G be a group, and S,T ⊆ G two distinct such that S ∩ T = ∅. If there is a homogeneous quasi-homomorphism ϕ : G → V such that either S ⊆ ker(ϕ) and T ⊆ G \ ker(ϕ), or S ⊆ G \ ker(ϕ) and T ⊆ ker(ϕ), CLASSIFICATION OF THE CONJUGACY CLASSES OF SLf 2(R) 13 then we say that S and T are separated by ϕ. Clearly, this notion only makes sense if

ϕ is homogeneous, otherwise one could take a bounded function such that ϕ(g) = 0V if g ∈ S and ϕ(g) = x 6= 0V if g ∈ T . A slightly weaker notion is the one of pointwise pairwise separation, which we define next.

Definition 1.19 (Quasi-separation). Two subsets S,T of a group G are said to be quasi-separated if, for each (s, t) ∈ S × T , there is a Banach space Vs,t (not necessarily equal for different pairs (s, t)) and a homogeneous quasi-homomorphism ϕs,t : G → Vs,t that separates {s} and {t}. If, in addition, there is a Banach space V such that one may assume Vs,t = V for every (s, t) ∈ S × T , then S and T are said to be quasi-separated via V. The next proposition shows that the usual notion of quasi-homomorphism (i.e., with codomain R) is sufficient to determine quasi-separation in general.

Proposition 1.20 (Universality of quasi-separation via R). Let G be a group, and S,T ⊆ G two distinct subsets. If S and T are quasi-separated, then they are quasi- separated via R.

Proof. It suffices to show that, given (s, t) ∈ S × T and some Banach space V, if there is a homogeneous quasi-homomorphism ϕ : G → V such that ϕ(s) = 0V and ϕ(t) 6= 0V, then there is a homogeneous quasi-homomorphism ψ : G → R such that ψ(s) = 0 and ψ(t) = 1. For that, let x := ϕ(t), and consider the linear functional f : X → R such that f(x) = 1, where X is the linear span of x. This linear functional is bounded, for |f(u)| = kxk−1 · kuk, for all u ∈ X. Hence, by the Hahn-Banach theorem (cf. Theorem 3.3, pp. 58–59 of Rudin [11]), there is a bounded linear −1 functional F : V → R such that F |X = f and |F (v)| ≤ kxk · kvk, for all v ∈ V. It only remains to check that ψ := F ◦ ϕ is a homogeneous quasi-homomorphism from G to R. The fact that ψ is a quasi-homomorphism follows from  |ψ(gh) − ψ(g) − ψ(h)| = F ϕ(gh) − ϕ(g) − ϕ(h) ≤ kxk−1 · kϕ(gh) − ϕ(g) − ϕ(h)k −1 ≤ kxk · Dϕ, where Dϕ is the defect of ϕ, and the homogeneity of ψ follows from the homogeneity of ϕ combined with the linearity of F . 

Finally, we state the main tool that will be used in the following sections.

Theorem 1.21. Let G be a group. If Z(G) \{e} is quasi-separated from the identity, then G is a CP-group.

Proof. By Lemma 1.18, any homogeneous quasi-homomorphism vanishes on the set Z(G) ∩ {[g, h] | g, h ∈ G}; thus, if Z(G) \{e} can be quasi-separated from {e}, which is always in the kernel (for k ϕ(e) = ϕ(ek) = ϕ(e), for every k ∈ Z), it follows that Z(G) ∩ {[g, h] | g, h ∈ G} = {e}. By Theorem 1.12 “(i) ⇐⇒ (iii)”, that implies that G  G/Z(G) is a CP-extension, meaning that G is a CP-group.  14 CHRISTIAN TAFULA´

1 2. The group Homeo] +(S ) 1 2.1. Homeo+(R) and Homeo+(S ). We say that a half-closed interval in R is posi- tively (resp. negatively) oriented if it is closed to the left (resp. to the right). Thus, for example, given x, y ∈ R, [x, y), [x, +∞) are positively oriented; (x, y], (−∞, y] are negatively oriented.

A similar notion can be defined for S1 := R/Z: a proper subset of S1 is called an oriented interval if it is the image of a half-closed interval by the natural quotient 1 map R  S , which we denote by x 7→ x, inheriting its orientation. Hence, for example, since 1 = 0, 1/3, 2/3, 2/3, 1/3 = [2/3, 4/3) are positively oriented; 1/3, 2/3, 2/3, 1/3 = (2/3, 4/3] are negatively oriented;

where U ⊆ S1 denotes the image of U ⊆ R by the natural quotient map. Notice that S1 = [x, y) t [y, x) for every x 6= y. Inside a fixed oriented interval, it makes sense to consider the order inherited from its pre-images, i.e., given z, w ∈ [x, y) resp. (x, y] we have ( z ≤ w if [x, z) ⊆ [x, w) resp. (z, y] ⊇ (w, y], z > w otherwise. Since taking boundaries commutes with homeomorphisms (i.e., ϕ(∂X) = ∂ϕ(X)), we deduce that, for any ϕ ∈ Homeo(S1) and any oriented interval I, the image ϕ(I) is again an oriented interval. If I and ϕ(I) have the same orientation, then we say that ϕ preserves the orientation of I, otherwise it reverses it. Finally, for x, y ∈ R, we also have a distance on S1 defined by d(x, y) := min |x − y − k|, k∈Z that is, the distance between x − y and its nearest integer. Equipped with this distance function, S1 becomes a compact metric space. Proposition 2.1. A homeomorphism ϕ ∈ Homeo(S1) either preserves all or reverses all the orientations of oriented intervals. Furthermore, 1  1 Homeo+(S ) := ϕ ∈ Homeo(S ) ϕ preserves orientation 1 1 is a subgroup of index 2 with Homeo(S )/Homeo+(S ) = {±idS1 }. 1 1  Proof. For ϕ ∈ Homeo(S ) and [x, y) (S , it holds either ϕ [x, y) = [ϕ(x), ϕ(y))  1 or ϕ [x, y) = (ϕ(y), ϕ(x)]. Fixing x 6= y ∈ S and letting z ≤ w ∈ [x, y), since ϕ[x, z) ⊆ ϕ[x, w) ⊆ ϕ[x, y), we conclude that ϕ must either preserve all or reverse all the orientations of [x, w) for w ≤ y. The same argument applies to [z, y) for z ≥ x, and thus, since [z, w) = [x, w) ∩ [z, y), CLASSIFICATION OF THE CONJUGACY CLASSES OF SLf 2(R) 15 it applies to all positively oriented subintervals of [x, y). By the bijectivity of ϕ, it follows immediately that the orientations of [x, y) and (x, y] are either both preserved or both reversed by ϕ. Hence, given any I ⊆ J (S1 oriented intervals, ϕ either preserves all or reverses all amongst I, J , S1 − I and S1 − J . Fixing J (S1, all oriented intervals must have a non-empty intersection with either J or S1 − J , and such intersection must contain some oriented interval. Therefore, if the orientation of J is preserved (resp. reversed) by ϕ, then the orientation of all oriented intervals is preserved (resp. reversed) as well. For the last statement, notice that the composition of two orientation reversing maps from Homeo(S1) is orientation preserving, and the inverse of an orientation preserving map (resp. reversing) is also orientation preserving (resp. reversing).  The situation in Homeo(R) is analogous, and for that reason, we omit the proof. Proposition 2.2. Every f ∈ Homeo(R) is strictly monotone. Furthermore,  Homeo+(R) := f ∈ Homeo(R) f is strictly increasing is a subgroup of index 2 with Homeo(R)/Homeo+(R) = {±idR}. 1 1 1 2.2. Universal covering of Homeo+(S ). The groups Homeo(S ) and Homeo+(S ) may be considered as quotients of certain subgroups of Homeo(R) and Homeo+(R), respectively. In order to see that, consider the following subgroups: 1  Homeo(] S ) := α ∈ Homeo(R) |α(x + 1) − α(x)| = 1 for all x ∈ R , 1  Homeo] +(S ) := α ∈ Homeo(R) α(x + 1) = α(x) + 1 for all x ∈ R . From continuity, we have that, for all α ∈ Homeo(] S1), either α(x + 1) = α(x) + 1 for all x ∈ R, or α(x + 1) = α(x) − 1 for all x ∈ R, 1 1 1 and thus, Homeo] (S )/Homeo] +(S ) = {±idR}. Each α ∈ Homeo] (S ) defines a self- homeomorphism α ∈ Homeo(S1) via α(x) := α(x), which is well-defined by construc- 1 tion. Amongst the elements of Homeo] +(S ), the simplest type consists of translations, which, for each u ∈ R, are defined as

(2.1) ∀x ∈ R, Tu(x) := x + u. 1 1 Since the kernel of Homeo] +(S ) 3 α 7→ α ∈ Homeo+(S ) clearly consists of the subgroup {Tk | k ∈ Z}, which is discrete, this map is a covering morphism. 1 1 Remark 2.3 (Metrizability). Both Homeo+(S ) and Homeo] +(S ) are metrizable. Indeed, the distance function on S1 allows us to define a topology on Homeo(S1) induced by the distance 1 ∀ϕ, ψ ∈ Homeo(S ), δsup(ϕ, ψ) := sup d(ϕ(x), ψ(x)) ∈ [0, 1/2], x∈S1 which can be shown to coincide with the compact-open topology. Using this distance, 1 one can show that Homeo+(S ) is a connected, locally path connected, locally simply connected Hausdorff topological group, and hence it admits a universal 16 CHRISTIAN TAFULA´ covering. The same may be deduced about Homeo+(R) with respect to the distance function

∀f, g ∈ Homeo+(R), ∆sup(f, g) := sup |f(x) − g(x)| ∈ R≥0 ∪ {+∞}, x∈R 1 as well as about the closed subgroup Homeo] +(S ). 1 1 The next two lemmas will be used for showing that Homeo] +(S )  Homeo+(S ) 1 is a CP-extension (Theorem 2.8). First and foremost, we show that Homeo] +(S ) is 1 indeed the universal covering of Homeo+(S ). 1 1 Lemma 2.4. Homeo] +(S ) is the universal covering group of Homeo+(S ). 1 Proof. We start by showing that the topological fundamental group of Homeo+(S ) 1  is isomorphic to Z, i.e., π1 Homeo+(S ) ' Z. Consider the subgroup  1 1 1 K := Ru : S 3 x 7→ x + u ∈ S u ∈ R ⊆ Homeo+(S ).

1 For any ϕ ∈ Homeo+(S ), let u := ϕ(0) and write ξ := R−u ϕ. This allows us to 1 write ϕ as a product of some Ru ∈ K and some ξ ∈ Homeo+(S ) that satisfies ξ(0) = 0. Hence, the map

1 1 Ψ : Homeo+(S ) × [0, 1] → Homeo+(S )

(Ru ξ, t) 7→ ξ(x) + u − t · (ξ(x) − x)

1 is well-defined and describes a strong retract from Homeo+(S ) to K. Thus, as ∼ 1 K 3 Ru 7−→ u ∈ S is an isomorphism of topological groups, we deduce that 1 1 π1(Homeo+(S )) ' π1(K) ' π1(S ), which is well-known to be isomorphic to Z. 1 Next, we need to show that Homeo] +(S ) is simply connected, i.e., that every loop 1 1 in Homeo] +(S ) is homotopically trivial. Let L : [0, 1] → Homeo] +(S ) be a loop, with 1 L(0) = L(1) = idR, the identity in R. Taking its image in Homeo+(S ), we have a 1 loop L, with L(0) = L(1) = idS1 . Using the strong retract from Homeo+(S ) to K, 1 we may assume that Im(L) ⊆ K. Thus, lifting it back to Homeo] +(S ) to another loop L0 with the same endpoints as L, we obtain a loop L0 which is homotopic to L and whose image consists of elements of the form Tu (i.e., liftings of Ru ∈ K). This 0 means that there is a f : [0, 1] → R such that L (t) = Tf(t), with f(0) = f(1) = 0. Hence, we may take a path of loops {γs | s ∈ [0, 1]} given by

Ls : [0, 1] 3 t 7→ T(1−s)f(t) ∈ Homeo] +(R) (for 0 ≤ s ≤ 1), 0 0 implying that L is homotopically trivial (for L0 = L and L1 is the identity loop). 1 We conclude that Homeo] +(S ) is simply connected, and therefore it is the universal 1 covering group of Homeo+(S ). 

1 The next lemma shows that the center of Homeo] +(S ) is generated by T1. 1  Lemma 2.5. Z Homeo] +(S ) = {Tk | k ∈ Z}' Z. CLASSIFICATION OF THE CONJUGACY CLASSES OF SLf 2(R) 17

Proof. The elements {Tk | k ∈ Z} are in the center by definition. To show the other 1  direction, take ζ ∈ Z Homeo] +(S ) . For each q ∈ N≥1 and x ∈ R, let sin(2πqx) α (x) := x + . q 2πq

It is easily checked that αq is strictly increasing, since its first derivative is never negative and has a discrete zero set; moreover, αq(x + 1) = αq(x) + 1 for all x ∈ R, 1 hence αq ∈ Homeo] +(S ). Since ζ is in the center, we have sin(2πζ(0)) ζ(α (0)) = α (ζ(0)) =⇒ ζ(0) = ζ(0) + 1 1 2π =⇒ sin(2πζ(0)) = 0

=⇒ ζ(0) ∈ Z.

Letting M := ζ(0), we are going to show that ζ = TM . For that, let ξ(x) := ζ(x) − M. It suffices to show that ξ is the identity when restricted to Q, for Q is dense in R and ξ is continuous. Fixing q and taking p ∈ {0, 1, . . . , q − 1}, we have sin(2πq ξ(p/q))  sin(2πq (p/q)) α ξ(p/q) = ξα (p/q) =⇒ ξ(p/q) + = ξ p/q + q q 2πq 2πq

=⇒ sin(2πq ξ(p/q)) = 0

=⇒ q ξ(p/q) ∈ Z.

Since ξ is strictly increasing, we have 0 ≤ ξ(p/q) < 1 for all 0 ≤ p < q ∈ Z≥1, and thus 0 < q ξ(1/q) < q ξ(2/q) < ··· < q ξ(1 − 1/q) < q ∈ Z, which implies q ξ(p/q) = p. From T1ξ = ξT1, we deduce that ξ(r) = r for all r ∈ Q, concluding the proof.  2.3. Translation number. Following Section 5 of Ghys [7], we define the translation 1 number of α ∈ Homeo] +(S ) as the limit αn(x) − x (2.2) τ(α) := lim , n→+∞ n where x ∈ R is arbitrary. The next two lemmas will show that τ is not only well- defined, but that it constitutes a homogeneous quasi-homomorphism. Thus, we will 1 1 deduce that Homeo] +(S )  Homeo+(S ) is a CP-extension by applying Theorem 1 1.21. Further invariants in Homeo] +(S ) will be discussed in Section 3. Lemma 2.6 (Well-definedness of τ). For each x ∈ R, consider the map 1 Ex :Homeo] +(S ) −→ R α 7−→ α(x) − x. Then, the following hold:

(i) For every x ∈ R, the map Ex is a quasi-homomorphism with defect at most 1. 18 CHRISTIAN TAFULA´

(ii) For every x, y ∈ , we have sup 1 |E (α) − E (α)| ≤ 1. R α∈Homeo^+(S ) x y n n (iii) For every x, y ∈ , we have lim Ex(α )/n = lim Ey(α )/n. R n→+∞ n→+∞ 1 (iv) τ :Homeo] +(S ) → R is a well-defined homogeneous quasi-homomorphism. 1 Proof. For any α, β ∈ Homeo] +(S ), we have   Ex(αβ) − Ex(α) − Ex(β) = α(β(x)) − β(x) − α(x) − x = α(T ) − T  − α(x) − x (2.3) = α(T ) − α(x) − T − x, where T := “unique t ∈ [x, x + 1) such that β(x) − t ∈ Z”. From continuity, we have α([x, x + 1]) = [α(x), α(x + 1)], and thus |α(T ) − α(x)| < 1. Since |T − x| < 1 and T − x has the same sign as α(T ) − α(x), it follows that 1 ∀α, β ∈ Homeo] +(S ), |Ex(αβ) − Ex(α) − Ex(β)| < 1; i.e., Ex is a quasi-homomorphism with defect at most 1, concluding item (i). For item (ii), note that calculating Ex(α) − Ey(α) yields an expression analogous to (2.3), from where it follows that sup 1 |E (α) − E (α)| ≤ 1. α∈Homeo^+(S ) x y n n Finally, since Ex(α ) = α (x) − x, items (iii) and (iv) follow from items (i), (ii) together with Lemmas 1.15, 1.16.  1 Lemma 2.7 (Basic properties of τ). Let α ∈ Homeo] +(S ). The following hold: (i) τ(Tu) = u for every u ∈ R. −1 1 (ii) τ(βαβ ) = τ(α) for every β ∈ Homeo] +(S ). (iii) τ(α) = 0 if and only if there is x∗ ∈ R such that α(x∗) = x∗. 1  3 (iv) Z Homeo] +(S ) \{idR} is separated from {idR} by τ. Proof. Item (i) is immediate from (2.2); item (ii) follows from Lemma 2.6 (iv) together with Lemma 1.17; and item (iv) follows from item (i), Lemma 2.5 and the definition of separation at the beginning of Subsection 1.5. For item (iii), note that, from Lemma 2.6 (iii), for every x, y ∈ R it holds αn(x) − x αn(y) − y τ(α) = lim = lim . n→+∞ n n→+∞ n Hence, if α has a fixed point x∗, it follows that αn(x∗) − x∗ = 0 for every n ≥ 1, and thus τ(α) = 0. For the converse, suppose that α has no fixed point. The function R 3 x 7→ α(x) − x ∈ R is a continuous periodic function invariant under translation by 1, and since α has no fixed points, α(x) − x is either positive everywhere or negative everywhere. Since [0, 1] is compact, there must be some ε > 0 such that |α(x) − x| ≥ ε > 0 for every x ∈ [0, 1], and consequently for every x ∈ R. That means that, for every x ∈ R and n ≥ 1, either αn(x) − x ≥ αn−1(x) − x + ε ≥ · · · ≥ α(x) − x + (n − 1)ε ≥ nε, or αn(x) − x ≤ αn−1(x) − x − ε ≤ · · · ≤ α(x) − x − (n − 1)ε ≤ −nε, depending on the sign of α(x) − x. In both cases, it follows that |τ(α)| ≥ ε > 0, implying the converse.  3Recall from Subsection 1.5 that separation is slightly stronger than quasi-separation. CLASSIFICATION OF THE CONJUGACY CLASSES OF SLf 2(R) 19

Armed with these properties, the main theorem of this section follows.

1 1 Theorem 2.8. Homeo] +(S )  Homeo+(S ) is a CP-extension. 1 Proof. From Lemma 2.7 (iv) and Theorem 1.21, we deduce that Homeo] +(S ) is a 1 CP-group, and since the center of Homeo] +(S ) is exactly the kernel of the universal 1 1 covering map Homeo] +(S )  Homeo+(S ) (Lemma 2.5), the theorem follows.  Consequently, all items from Theorem 1.12 apply to this extension. Remark 2.9 (Counterexample to the converse of Remark 1.13 (iii)). Theorem 2.8 1 1 says that Homeo] +(S )  Homeo+(S ) is a CP-extension, and Lemma 2.5 implies 1  1 1  that Z Homeo] +(S ) ' Z is torsion-free. However, Homeo] +(S )/Z Homeo] +(S ) ' 1 1 1 Homeo+(S ) is clearly not a torsion group (e.g., consider Ru : S 3 x 7→ x + u ∈ S for some u ∈ [0, 1) \ Q). 2.4. Canonical liftings. For practical purposes, in order to apply Theorem 1.12 (v) 1 on subgroups of Homeo+(S ) (in view of Lemma 1.4 (iii)), it helps to have a fixed set 1 1 of liftings of Homeo] +(S )  Homeo+(S ). Certain groups G possess a canonical class of representatives for inequivalent conjugacy classes (such as the Jordan normal form in GLn(C)), thus allowing one to write the set of conjugacy classes of a CP-extension Ge  G explicitly and canonically. The next proposition constructs a consistent 1 choice of such liftings for Homeo] +(S ). Proposition 2.10 (Canonical liftings). The rows and columns of the following commutative diagram are exact: 1 1

k 7→ Tk 1 1 1 Z Homeo] +(S ) Homeo+(S ) 1

k 7→ Tk 1 Z Homeo(] S1) Homeo(S1) 1   0 if increasing, 0 if preserves oriented intervals, α 7→ ϕ 7→ 1 if decreasing. 1 if reverses it. Z/2Z Z/2Z

1 1

Proof. If α ∈ Homeo] (S1) is increasing, then its image α ∈ Homeo(S1) is clearly orientation preserving, and thus, the exactness of the columns and commutativity of the middle square follow from Proposition 2.1. For the rows, it is clear that Tn = idS1 for all n ∈ Z. Hence, all we need to show is that α 7→ α is surjective and that 1 α = idS1 implies α ∈ hT1i. In order to do so, we construct a lifting ϕe ∈ Homeo] (S ) 20 CHRISTIAN TAFULA´ of ϕ ∈ Homeo(S1) which is canonical, in the sense of being the unique family of 1 liftings satisfying ϕe(0) ∈ [0, 1) and (]−ϕ) = −ϕe for every ϕ ∈ Homeo+(S ). If ϕ is orientation preserving (resp. reversing), we define  ϕe(0) := the unique real number in [0, 1) resp. (−1, 0] such that ϕe(0) = ϕ(0). We also define, for x ∈ [0, 1),  ϕe(x) := the unique lifting of ϕ(x) in [ϕe(0), ϕe(0) + 1) resp. (ϕe(0) − 1, ϕe(0)] , and, for any n ∈ Z, ϕe(x + n) := ϕe(x) + n (resp. ϕe(x) − n). The following two assertions yield our theorem: 1 1 1 • Assertion 1: ϕe ∈ Homeo] +(S ) (resp. Homeo(] S ) \ Homeo] +(S )). For any 0 ≤ t < 1, we have ϕ[0, t) = [ϕ(0), ϕ(t)) (resp. (ϕ(t), ϕ(0)]), hence  ϕe [0, t) = [ϕe(0), ϕe(t)) (resp. (ϕe(t), ϕe(0)]), implying that ϕe|[0,1) is bijective, bi- continuous, and increasing (resp. decreasing). Since  ϕe(x) = bxc + ϕe|[0,1)(x − bxc) resp. − bxc + ϕe|[0,1)(x − bxc) for all x ∈ R, it is clear that ϕe is also bijective and bi-continuous. • Assertion 2: If α = ϕ, then α = Tn ϕe for some n ∈ Z. For 0 ≤ t < 1, we have α(t) = α(t) = ϕ(t); thus, by construction, ϕe(t) and α(t) have the same fractional part, i.e., α(x) − ϕe(x) is an integer, for every x ∈ R. Since both α and ϕ are continuous and Z is discrete, we conclude that α(x) − ϕ(x) is e −1 e constant, and thus αϕe ∈ hT1i.  Whenever we refer to Proposition 2.10, we will be referring to the construction of the 1 canonical liftings, and to the fact that two distinct liftings of some ϕ ∈ Homeo+(S ) differ by a central element (cf. Lemma 2.5). The following figure shows an example to illustrate how canonical liftings look like:

-2 -1 0 1 2 ◦ ◦ •◦ ◦ ◦ R

ϕe

◦ ◦ •◦ ◦ R -2 -1 0 ϕe(0) 1 ϕe(1) 2

Figure 1. Canonical lifting of ϕ : S1 3 x 7→ x + 1/2 ∈ S1.

Remark. On Lemma 3.5, p. 150 of Zhang [12], a different lifting is considered for the 1 subgroup SLf2(R) ⊆ Homeo] +(S ), which is yet to be defined in this paper. The lifting considered there is uniquely determined as well, hence called canonical. However, we work with a different canonical lifting in general, whose construction resorts to no particular property of linear groups. See Remark 4.4 for details. CLASSIFICATION OF THE CONJUGACY CLASSES OF SLf 2(R) 21

1 1 Consider now a subgroup G ⊆ Homeo+(S ), and define Ge ⊆ Homeo] +(S ) to be the group that makes the following diagram commute, whilst having exact rows:

1 Ge G 1

Z' p (2.4) 1  Z Homeo] +(S ) '

1 1 1 hT1i Homeo] +(S ) Homeo+(S ) 1.

By Theorem 2.8 and Lemma 1.4 (iii), Ge  G is a CP-extension. Write I := C`(G), and for each i ∈ I, choose and fix a representative element ϕi ∈ G, so that C`(G) = {[ϕi]G | i ∈ I}. It is clear that the canonical liftings of elements of G, in the sense of Proposition 2.10, are elements of Ge. Thus, the following is an immediate consequence of Theorem 1.12 “(i) ⇐⇒ (iv)” and Proposition 1.8:

Corollary 2.11. For every α ∈ Ge, there is a unique i ∈ I (as defined above) and a unique k ∈ for which α ∼ T ϕ . Z Ge k ei 1 3. Invariants in Homeo] +(S ) The goal of this section is to describe three invariants associated to an element 1 α ∈ Homeo] +(S ), these being: • The translation magnitude |τ(α)| ∈ R≥0. • The translation direction σ(α) ∈ {−1, 0, 1}. ] ] • The length-sharp ` (α) ∈ R≥[0]. 1 All three notions encode some way to “displacement” in Homeo] +(S ). The translation magnitude |τ| is just the of the translation number τ, defined in (2.2). The translation direction σ, on the other hand, is a refinement of the sign of τ, so that the pair (|τ|, σ) carries more information than just τ. This pair gives us the direction type (cf. Definition 3.1). The length-sharp `] is based on the notion of length (denoted by `) defined in Lemma 3.7 of Zhang [12]. Conceptually speaking, the function `] may be conceived of as a conjugacy invariant version of `, based on the “δsup” function defined in Section 3 of Mochizuki [9]. In the sense of measuring how far an element from 1 α ∈ Homeo] +(S ) is from the identity idR, ` may be understood as sharper than τ, whilst `] is a coarser version of `, capturing the same phenomenon modulo 1 conjugacy equivalence. Although we focus solely on Homeo] +(S ), some elements of n our discussion can naturally be extended to further Homeo] 0(T ). 3.1. Direction type. The translation number function τ does not differentiate 1 between the α ∈ Homeo] +(S ) which have a fixed point (cf. Lemma 2.7 (iii)). Thus, 1 for this purpose, we introduce the translation direction σ. For α ∈ Homeo] +(S ), we define σ(α) as follows: • If |τ(α)| > 0, then σ(α) := τ(α)/|τ(α)|;

• If α 6= idR, τ(α) = 0 and τ(Tε α) > 0 for every ε > 0, then σ(α) := +1; • If α 6= idR, τ(α) = 0 and τ(T−ε α) < 0 for every ε > 0, then σ(α) := −1; • If none of the above is satisfied, we define σ(α) := 0. 22 CHRISTIAN TAFULA´

The case where both τ(Tε α) > 0 and τ(T−ε α) < 0 hold for all ε > 0 is only satisfied by the identity. Indeed, if α =6 idR, then there is x ∈ R such that either α(x) > x or α(x) < x; thus, letting ε := |α(x) − x|, either τ(T−ε α) = 0 or τ(Tε α) = 0. Therefore, 1 σ is well-defined for all elements of Homeo] +(S ). With that, consider the following: 1 Definition 3.1 (Direction types). Given α ∈ Homeo] +(S ) with α 6= idS1 , the direction type of α is defined as one of the following five possibilities: Direction type (|τ|, σ)-pair Description

Forward (+, +) ∀x ∈ R, α(x) − x > 0 Semi-forward (0, +) α is fixed; ∀x ∈ R, α(x) − x ≥ 0 Alternating (0, 0) α(x) − x changes sign

Semi-backward (0, −) α is fixed; ∀x ∈ R, α(x) − x ≤ 0 Backward (+, −) ∀x ∈ R, α(x) − x < 0

If α = idS1 , then we simply say it is the identity. In case α has forward direction type (resp. backward), we say that α moves forward (resp. backward), and if it has semi-forward direction type (resp. semi-backward), then we say that α is semi-forward (resp. semi-backward). If α has alternating direction type, then so does α−1. However, if α moves forward (resp. is semi-forward), then α−1 moves backward (resp. is semi-backward); that is, the direction type of the inverse of an element is the inverse of the direction type of that element. Since |τ| is just the absolute value of a homogeneous quasi-homomorphism, it follows immediately from Lemma 1.17 that it is invariant under conjugation. The same holds for σ, which follows from the following lemma: 1 Lemma 3.2. Direction type is invariant under conjugation in Homeo] +(S ).

Proof. Let α, β ∈ Homeo] +(R/Z). If α moves forward, then α(x) > x for all x ∈ R, which implies α(β−1(x)) > β−1(x); thus, as ββ−1(x) = x, it follows that βαβ−1(x) > x. The same holds for semi-forward case, and the case where α moves backward or is semi-backward goes analogously. If α has a fixed point, say x∗ ∈ R, then β(x∗) is a fixed point of βαβ−1, which implies that having a fixed point is invariant under conjugation. Since the alternating case is the only case with fixed points left, it has to be conjugacy invariant too.  The specific value of τ(α) will not be too important for us during the remainder of this paper. Instead, we shall make more use of the length-sharp function as a measure of how much α “moves”. Hence, the two invariants mentioned at the beginning of this section (namely, |τ| and σ) are more accurately conceptualized in our context as simply the parts that constitute the direction type. 3.2. Displacement length. Consider now the length function: 1 (3.1) ` :Homeo] +(S ) → R≥0, `(α) := sup |α(t) − t|. t∈[0,1] Since α(t + 1) = α(t) + 1 for all t ∈ R and [0, 1] is compact, ` is well-defined, and we have `(α) = maxt∈[x,x+1] |α(t) − t| for all x ∈ R, with `(α) = 0 if and only if α = idR. CLASSIFICATION OF THE CONJUGACY CLASSES OF SLf 2(R) 23

1 ◦ 1 ◦ 1 ◦

.5 .5 .5

• • • 0 .5 1 0 .5 1 0 .5 1

sin(2πx) 1 sin(2πx) 1 α(x) = x + 2π + π α(x) = x α(x) = x + 2π − π (Forward) (Identity) (Backward)

◦ ◦ ◦ 1 1 × 1 × .5 .5 × .5 × • • • 0 .5 1 0 × .5 1 0 .5 1

sin(2πx) 1 sin(2πx) sin(2πx) 1 α(x) = x + 2π + 2π α(x) = x + 2π α(x) = x + 2π − 2π (Semi-forward) (Alternating) (Semi-backward)

Figure 2. Illustrations of the graphs of example functions for each possible direction type. The functions, besides the identity, are just variations of the function α1 used in the proof of Lemma 2.5.

Recalling the distance δsup introduced in Remark 2.3, it follows directly from the triangle inequality that ` is 1-Lipschitz, and hence continuous. Moreover, as one would expect for a notion intended to measure how far α is from the identity, for any translation Ty with y ∈ R it holds `(Ty) = |y|. 1 Lemma 3.3 (Basic properties of `). Let α, β ∈ Homeo] +(S ). The following hold: (i) |τ(α)| ≤ `(α); (ii) `(αβ) ≤ `(α) + `(β); (iii) If α has a fixed point, then `(α) < 1; (iv) `([α, β]) < 2. Proof. For item (i), for each n ≥ 1 we have |αn(x) − x| ≤ |α(αn−1(x)) − αn−1(x)| + |αn−1(x) − x| ≤ `(α) + |αn−1(x) − x|, and thus, |αn(x) − x| ≤ n `(α), which implies |τ(α)| ≤ `(α). For item (ii), since β is bijective and β(1) = β(0) + 1, we have `(αβ) = max |αβ(t) − t| t∈[0,1] 24 CHRISTIAN TAFULA´

≤ max |αβ(t) − β(t)| + |β(t) − t| t∈[0,1] ≤ max |α(u) − u| + max |β(t) − t| = `(α) + `(β). u∈[β(0),β(1)] t∈[0,1] ∗ For item (iii), let x ∈ R be some fixed point of α. Since `(α) = supt∈[x∗,x∗+1] |α(t)− t|, take t∗ ∈ [x∗, x∗ + 1] such that `(α) = |α(t∗) − t∗|. From the fact that α is strictly increasing and α(x∗ + 1) = α(x∗) + 1, we deduce that α([x∗, x∗ + 1]) = [x∗, x∗ + 1]. Therefore, we have T ∗ := α(t∗) ∈ (x∗, x∗ + 1), the endpoints of the interval being excluded because α(x∗) = x∗, from where it follows that `(α) = |T ∗ − t∗| < 1. 1 For item (iv), by Proposition 2.10, there are ϕ ∈ Homeo+(S ) and n ∈ Z such that α = Tnϕe. Since Tn is an element of the center, it follows that [α, β] = [ϕ,e β] = ϕβϕ−1β−1. From Lemma 3.2, we know that ϕ−1 and βϕ−1β−1 have the same direction e e −1 −1e e type, therefore, either both ϕe and βϕe β have fixed points or they have opposite directions (i.e., one moves forward and the other one backward). We consider these two cases separately: Case 1: If both ϕ and βϕ−1β−1 have fixed points, then by item (iii) we have `(ϕ) < 1 −e1 −1 e e and `(βϕe β ) < 1. Thus, by item (ii), we conclude that `([ϕ,e β]) < 2. Case 2: Suppose that ϕe moves forward. By the construction in Proposition 2.10, we have 0 ≤ ϕe(0) < 1, hence, maxt∈[0,1] |ϕe(t) − t| ≤ |ϕe(1) − 0| < 2. Since β is a strictly increasing self-homeomorphism of R, it holds −1 ϕe (t) > t − 2 (applying β on both sides) −1 −1 =⇒ βϕe (t) > β(t) − 2 (substituting t 7→ β (u)) −1 −1 =⇒ βϕe β (u) > u − 2, −1 −1 −1 −1 which implies that `(βϕe β ) < 2 as well. Hence, since βϕe β moves backward, for all t ∈ R we have −1 −1  −1 −1 0 ≤ ϕe βϕe β (t) − βϕe β (t) < 2, −1 −1 −2 < βϕe β (t) − t ≤ 0, and thus −1 −1 −2 < ϕβe ϕe β (t) − t < 2, which is equivalent to `([ϕ,e β]) < 2, as required. The case for when ϕe moves backward goes analogously.  In order to define a conjugacy invariant notion of displacement length, we have to allow for certain indeterminacies, because ` is not invariant under conjugation in general. To be more precise, consider ( ] ∼ x = y ∈ Z, or R := R /∼=, where x = y ⇐⇒ x, y ∈ (k, k + 1) for some k ∈ Z. = {..., [(−2, −1)], [−1], [(−1, 0)], [0], [(0, 1)], [1], [(1, 2)],...}, and denote the quotient map by R 3 x 7→ x] ∈ R]. This object also arises as the 1 quotient of R through the action of the subgroup of Homeo] +(S ) which fixes Z, i.e., Stab 1 (0) , where Homeo^+(S ) y R 1 Stab 1 (0) := {α ∈ Homeo ( ) | α(0) = 0}. Homeo^+(S ) ] + S CLASSIFICATION OF THE CONJUGACY CLASSES OF SLf 2(R) 25

] 1 Despite the seemingly natural identification of R with the half-integers 2 Z through the obvious order isomorphism, the inherent group structure of the latter is not shared by the former, hence, since the order-preserving property of x 7→ x] is our only concern, we avoid such identification. A more appropriate alternative description of x] would be the set {bxc, dxe}, which highlights the precise loss of information that takes place with mapping x 7→ x]; such information loss is what we call indeterminacies. ] ] ] The main feature of this quotient is that, defining ` (α) := `(α) ∈ R , we have ] 1 Proposition 3.4. ` is invariant under conjugation in Homeo] +(S ). 1 Proof. Let α, β ∈ Homeo] +(S ). If α 6= idS1 has a fixed point, then by Lemma 3.3 ] −1 (iii) we have ` (α) = [(0, 1)], and by Lemma 3.2 we have that βαβ (6= idS1 ) has a fixed point as well, implying that `](βαβ−1) = `](α) = [(0, 1)]. Assume, then, that α moves forward. We are going to show that `(α) < n =⇒ `(βαβ−1) < n, `(α) ≤ n =⇒ `(βαβ−1) ≤ n, `(α) > n =⇒ `(βαβ−1) > n and `(α) ≥ n =⇒ `(βαβ−1) ≥ n. If `(α) < n, suppose that `(βαβ−1) ≥ n; i.e., that there is x ∈ R such that βαβ−1(x) ≥ x + n. −1 −1 −1 Since β is increasing and it commutes with Tn, we have α(β (x)) ≥ β (x) + n, which implies that there is y ∈ R for which α(y) − y ≥ n, leading to contradiction. The case for when `(α) ≤ n follows from the exact same argument. If `(α) ≥ n, suppose that `(βαβ−1) < n; i.e., that for every x ∈ R we have βαβ−1(x) < x + n. −1 −1 −1 Since β is increasing and it commutes with Tn, we have α(β (x)) < β (x) + n, which implies that for every y ∈ R we have α(y) − y < n, leading to contradiction. The same follows for when `(α) > n, and the case for when α moving backward goes analogously, concluding the proof. 

4. Conjugacy classes of SLf2(R)

We now finally turn our attention to PSL2(R). In order to apply the results already discussed, our first goal will be to describe a homomorphism of topological groups 1 PSL2(R) ,→ Homeo+(S ) that is a homeomorphism onto its image, i.e., a topological embedding. This is done with the aid of the following lemma. Lemma 4.1. Let X,Y,Z be topological spaces, with Y locally compact Hausdorff, and let S ⊆ C(Y,Z) be a set of continuous maps endowed with the compact-open topology. Given a map φ : X → S, the following are equivalent:

(i) φ : X 3 x 7→ φx ∈ S is continuous; (ii) The map Φ: X × Y 3 (x, y) 7→ φx(y) ∈ Z is continuous.

Proof. The topology on S is generated by the sets VK,U := {f ∈ S | f(K) ⊆ U}, for every K ⊆ Y compact and U ⊆ Z open, hence, the map φ is continuous if and only if the sets VK,U := {x ∈ X | φx(K) ⊆ U} ⊆ X are open. −1 Now, consider Φ (U) = {(x, y) | φx(y) ∈ U} ⊆ X × Y , and take (x, y) ∈ −1 Φ (U). Since φx is continuous and Y is locally compact Hausdorff, y has an open 26 CHRISTIAN TAFULA´

−1 neighborhood Ny such that its N y is compact and y ∈ Ny ⊆ N y ⊆ φx (U). −1 Thus, (x, y) ∈ VN y,U × Ny ⊆ Φ (U), from where it is clear that (i) =⇒ (ii). For the other direction, fix VK,U ⊆ X and take x ∈ VK,U . If Φ is continuous, then (y) for every y ∈ K there are open neighborhoods Wx 3 x (in X) and Ny 3 y (in Y ) (y) −1 such that Wx × Ny ⊆ Φ (U). Since {Ny | y ∈ K} covers K, which is compact, we

may take a finite subcovering Ny1 ,...,Nyn , with n ∈ N and yi ∈ K, that covers K as Tn (yi) well. Thus, take Wx := i=1 Wx , which is a finite intersection of open sets, hence open. It follows that x ∈ Wx ⊆ VK,U , therefore VK,U is open and (ii) =⇒ (i).  2 In our case, the natural action of SL2(R) on R gives us a map 2 2 SL2(R) × R \{(0, 0)} 3 (M, v) 7−→ Mv ∈ R \{(0, 0)} which is obviously continuous, hence, the induced map on the quotient (R2 \ {(0, 0)})/R ' S1 is continuous as well. This is well-defined, because matrices in SL2(R) take 1-dimensional subspaces to 1-dimensional subspaces. Notice that this quotient is not the circle in R2, but the unit circle modulo a reflection through the origin (i.e., the real projective line), which is then identified with R/Z, and 1 hence S . Thus, as this induced map factors through PSL2(R), Lemma 4.1 gives us 1 continuous map PSL2(R) → Homeo+(S ), which amounts to a continuous action 1 PSL2(R) y S . The fact that this action is faithful follows from the simple fact that if M,N ∈ SL2(R) map 1-dimensional subspaces to the same 1-dimensional subspace, −1 λ 0 then MN is a homothety from the origin, i.e., of the form ( 0 λ ) for some λ ∈ R. −1 Since det(MN ) = 1, that implies λ = ±1, and thus M = N ∈ PSL2(R). It follows 1 that the map PSL2(R) ,→ Homeo+(S ) is then an embedding of topological groups.

4.1. Universal covering of PSL2(R). Initially, start by letting SLf2(R) denote the subgroup of self-homeomorphisms of R which induce an element of PSL2(R). In 1 other words, we define SLf2(R) ⊆ Homeo] +(S ) to be the group Ge in the diagram at (2.4), with G = PSL2(R). The fact that this group is the actual universal covering group of PSL2(R) follows as consequence of the Iwasawa decomposition of PSL2(R), also known as KAN decomposition. Considering the subgroups  cos(ϑ) − sin(ϑ)  K := ρ := 0 ≤ ϑ < π , ϑ sin(ϑ) cos(ϑ)  λ 0   A := a := λ > 0 , λ 0 1/λ  1 x  N := u := x ∈ , x 0 1 R the following holds:

a b Lemma 4.2 (Iwasawa decomposition). Every [ c d ] ∈ PSL2(R) has a unique repre- a b sentation as [ c d ] = ρϑaλux, where ρϑ ∈ K, aλ ∈ A, and ux ∈ N.

Proof. We follow the proof presented in K. Conrad [3]. Consider the action of PSL2(R) on the upper half-plane h := {z ∈ C : Im(z) > 0}, given by a b az + b (z) = . c d cz + d CLASSIFICATION OF THE CONJUGACY CLASSES OF SLf 2(R) 27

A routine calculation shows that this action is well-defined and faithful. To see that it is transitive, notice that for any x + iy ∈ h we have √ √  y x/ y u a√ (i) = √ (i) = x + iy, x y 0 1/ y therefore there can only be one orbit. Furthermore, since uxaλ = aλux/λ2 we have that NA = AN. Lastly, since ( a b a = d, (i) = i ⇐⇒ =⇒ a2 + c2 = 1, c d b = −c,

a b a b one readily sees that Stab(i) := {[ c d ] ∈ PSL2(R) | [ c d ](i) = i} = K. Hence, any a b matrix [ c d ] ∈ PSL2(R) may be written as an element of K times an element of AN a b which is determined by [ c d ](i), yielding the decomposition PSL2(R) = KAN. 

Similar to what was done in Corollary 2.4, we may show that π1(PSL2(R)) ' Z by considering the map

φ : PSL2(R) × [0, 1] → PSL2(R) cos(ϑ) − sin(ϑ) λt 0  1 tx (ρ a u , t) 7→ ρ a t u = ϑ λ x ϑ λ tx sin(ϑ) cos(ϑ) 0 λ−t 0 1 which is a strong retract from φ(PSL2(R)×{1}) = PSL2(R) to φ(PSL2(R)×{0}) = K. 1 The identification of PSL2(R) with its isomorphic image in Homeo+(S ) also identifies the K described by the Iwasawa decomposition with the  1 ∼ 1 K = ρϑ : S 3 x 7−→ x + ϑ/π ∈ S 0 ≤ ϑ < π 1 used on the proof of Corollary 2.4, and thus π1(PSL2(R)) ' π1(K) ' π1(S ) ' Z. The remaining part of the argument to show that SLf2(R) is the universal covering 1 group of PSL2(R) is entirely similar to one used for Homeo+(S ), with emphasis to the fact that all the liftings of loops in PSL2(R) are in SLf2(R), by definition.

• ◦ × • ×◦ • ×◦

K A N

Figure 3. Illustration of how generic elements from each of the groups 1 from the Iwasawa decomposition of PSL2(R) act on S . The circles represent S1 in the usual counter-clockwise orientation; the arrows indicate, roughly, the direction and length of the rate of change in the sector below it; and “×” indicates fixed points. 28 CHRISTIAN TAFULA´

Remark 4.3 (Center of SLf2(R)). We claim that Z(PSL2(R)) = {Id}. Indeed, suppose a b 1 x a b 1 −x a b that [ c d ] ∈ Z(PSL2(R)). Then, [ 0 1 ][ c d ][ 0 1 ] = [ c d ] for every x ∈ R, implying that  λ 0  1 b  1/λ 0  1 b c = 0 and a = d = ±1. Moreover, since 0 1/λ [ 0 1 ] 0 λ = [ 0 1 ] for every λ ∈ R≥1, a b 1 0 it follows that b = 0, and hence [ c d ] = [ 0 1 ] = Id, implying that Z(PSL2(R)) is trivial. Consequently, the only central elements of SLf2(R) are the liftings of the identity. Hence, through the identification of SLf2(R) as a topological subgroup of 1  1  Homeo] +(S ), we have Z SLf2(R) = Z Homeo] +(S ) . Remark 4.4 (Alternative liftings). In Lemma 3.5, p. 150 of Zhang [12], a different section of the covering SLf2(R)  PSL2(R) is considered; i.e., not the canonical liftings we established in Proposition 2.10. The construction of such alternative liftings may be briefly reproduced as follows. For an element γ ∈ PSL2(R), there are two possibilities: (i) Fixed: If γ has a real eigenspace,4 then it is clear that its induced self- 1 homomorphism ϕγ ∈ Homeo+(S ) has a fixed point. Thus, one takes γb ∈ SLf2(R) to be the unique lifting of ϕγ which has a fixed point. a b T 2 (ii) General case: Let γ = [ c d ]. If c = 0, then (1, 0) ∈ R is an eigenvector of a b ( c d ) ∈ SL2(R), and thus it falls into the previous case. Hence, we may assume c 6= 0. With that, we have a b 1 x (a − 1)(d − 1) det − = 0, where x := b − , c d 0 1 c

a b 1 x −1 −1 and thus, det(( c d )( 0 1 ) − Id) = 0, implying that ux, γux ∈ PSL2(R) have −1 each a real eigenspace, falling into the previous case. Writing γ1 := γux , we may then define γb := γb1bux. Notice that x = x(γ) is the unique real number satisfying the det = 0 condition, therefore γb ∈ SLf2(R) is uniquely determined.

4.2. Conjugacy classes. In order to write the conjugacy classes of SLf2(R) explicitly, we first need to write those of PSL2(R). This is a well-known classification, and we include a short proof for the sake of completeness.

1 0 a b Lemma 4.5. Write Id := [ 0 1 ] and let [ c d ] ∈ PSL2(R). In the notation of Lemma 4.2, the following hold: a b a b • If tr([ c d ]) > 2, then there is a unique λ > 1 for which [ c d ] ∼ aλ; a b a b • If tr([ c d ]) = 2, then [ c d ] is conjugate to either Id, u1 or u−1; a b a b • If tr([ c d ]) < 2, then there is a unique 0 < ϑ < π for which [ c d ] ∼ ρϑ. a b 2 a b Proof. The characteristic polynomial of ( c d ) ∈ SL2(R) is given by x −tr(( c d ))x+1, a b whose roots are the eigenvalues of ( c d ). The nature of such eigenvalues is determined a b 2 by the sign of the discriminant tr(( c d )) − 4 of this polynomial as follows: if a b 2 a b tr(( c d )) > 4, then ( c d ) has two distinct real eigenvalues, say λ and 1/λ for some a b 2 non-zero λ ∈ R; if tr(( c d )) = 4 then its eigenvalues are either both 1 or both −1; a b 2 iϑ and, if tr(( c d )) < 4, then it has a pair of complex conjugate eigenvalues, say e and e−iϑ for some 0 < ϑ < π.

4  a b  a b  −a −b  Meaning that, if γ = c d , then either c d or −c −d ∈ SL2(R) has a real positive eigenvalue. CLASSIFICATION OF THE CONJUGACY CLASSES OF SLf 2(R) 29

a b In case ( c d ) has two linearly independent real eigenvectors, the conjugating matrix may be assumed to be in SL2(R), as it can be taken to be the matrix formed by a basis of normalized eigenvectors. From the det = 1 condition and the quotient by {±Id}, we deduce that a b λ 0  ∼ , for some λ > 0. c d 0 1/λ

0 1  λ 0  0 1 −1  1/λ 0  If λ = 1, this covers the Id case; otherwise, since [ −1 0 ] 0 1/λ [ −1 0 ] = 0 λ , we may always assume λ > 1, which covers the case tr > 2. a b In case ( c d ) has repeated eigenvalues but a real eigenspace of dimension 1, its eigenvalues must be either both 1 or both −1. Taking a normalized eigenvector and extending it to a positively oriented orthonormal basis, the matrix formed by this basis is in SL2(R), and hence a b 1 x ∼ , for some non-zero x ∈ . c d 0 1 R

 ε f  1 x  ε f −1 h 1−εgx ε2x i −1/2 Since g h [ 0 1 ] g h = −g2x 1+εgx , taking g = 0 and ε = |x| =6 0 implies a b a b 2 that either [ c d ] ∼ u1 or [ c d ] ∼ u−1. Finally, since ε > 0, we have u1 6∼ u−1. a b iϑ −iϑ Lastly, if ( c d ) has a pair of complex conjugate eigenvalues e and e , consider z z 2 a b iϑ a pair of eigenvectors u = ( w ), v = ( w ) ∈ C , with ( c d )u = e u. Since 2 Re(z) 2 Im(z) u + v = , i(u − v) = ∈ 2 2 Re(w) 2 Im(w) R are linearly independent, and a b (u + v) = cos(ϑ)(u + v) + sin(ϑ)i(u − v), c d a b i(u − v) = − sin(ϑ)(u + v) + cos(ϑ)i(u − v), c d

−1  2Re(z) 2Im(z)  a b  2Re(z) 2Im(z)   cos(ϑ) − sin(ϑ)  we have 2Re(w) 2Im(w) ( c d ) 2Re(w) 2Im(w) = sin(ϑ) cos(ϑ) . Scaling u and v by some real number and swapping its values if necessary, we may guarantee that this conjugating matrix has det = 1. Furthermore, since cos(ϑ + π) = − cos(ϑ) and sin(ϑ + π) = − sin(ϑ), we may restrict ϑ to the interval (0, π), and thus a b cos(ϑ) − sin(ϑ) ∼ = ρ , for some 0 < ϑ < π, c d sin(ϑ) cos(ϑ) ϑ concluding the last remaining case. 

Denoting by ρeϑ, eaλ, eux the canonical liftings of ρϑ, aλ, ux in the sense of Proposition 2.10, we arrive at the following:

Theorem 4.6. In the notation of Lemma 4.5, the conjugacy classes of SLf2(R) are described by the following table, with each ϑ ∈ (0, π), λ ∈ R>1, and n ∈ Z≥1 denoting a different conjugacy class: 30 CHRISTIAN TAFULA´

Conjugacy class Length-sharp (`])

[idR] [0] [eu1], [eu−1], [eaλ], [ρeϑ], [T−1 ρeϑ] [(0, 1)] [Tn], [T−n], [Tn eu1], [T−n eu−1][n]

[T−n u1], [Tn u−1], [Tn aλ] e e e [(n, n + 1)] [T−n eaλ], [Tn ρeϑ], [T−1−n ρeϑ] Proof. By Corollary 2.11 and Lemma 4.5, we know that the conjugacy classes listed above are all inequivalent and represent all of SLf2(R). Hence, it only remains for us to check the length-sharp of each class, which we know by Proposition 3.4 to be conjugacy invariant. ] ] ] Clearly, ` (idR) = [0] and ` (Tn) = ` (T−n) = [n]. Furthermore, since ρeϑ = Tϑ/π, ] ] for every k ∈ Z≥0 we have ` (Tk ρeϑ) = ` (T−1−k ρeϑ) = [(k, k + 1)]. Since eu1, eu−1 and ] ] ] aλ have fixed points, by Lemma 3.3 (iii) it holds ` (u1) = ` (u−1) = ` (aλ) = [(0, 1)]. e 1 e e e From the definition of the action PSL2(R) y S , one may verify that eu−1 is has semi-forward direction whilst eu1 is semi-backward. From Lemma 3.3 (iii), we know that `(eu−1), `(eu1) < 1, and thus it follows that, for every n ≥ 1, ] ] ` (Tn eu1) = ` (T−n eu−1) = [n], ] ] ` (T−n eu1) = ` (Tn eu−1) = [(n, n + 1)]. ] Finally, when it comes to aλ, since `(aλ) < 1 and aλ is alternating, both ` (Tn aλ) ] e e e e and ` (T−n eaλ) must be equal to [(n, n + 1)], thus completing the proof. 

Direction Length- Conjugacy Translation Trace type sharp (`]) class number (τ)

[(0, 1)] [ρϑ] ϑ/π Forward e [(n, n + 1)] [Tn ρϑ] ϑ/π + n tr < 2 e [(0, 1)] [T−1 ρϑ] ϑ/π − 1 Backward e [(n, n + 1)] [T−1−n ρeϑ] ϑ/π − (n + 1)

[n][Tn u1] n Forward e [(n, n + 1)] [Tn eu−1] n tr = 2 Semi-forward [(0, 1)] [eu−1] 0

(α 6= idS1 ) Semi-backward [(0, 1)] [eu1] 0

[n][T−n u−1] −n Backward e [(n, n + 1)] [T−n eu1] −n CLASSIFICATION OF THE CONJUGACY CLASSES OF SLf 2(R) 31

Direction Length- Conjugacy Translation Trace type sharp (`]) class number (τ)

Forward [n][Tn] n α = id 1 S Identity [0] [id ] 0 (α central) R Backward [n][T−n] −n

Forward [(n, n + 1)] [Tn eaλ] n tr > 2 Alternating [(0, 1)] [eaλ] 0 Backward [(n, n + 1)] [T−n eaλ] −n

As a consequence of Theorem 4.6, we deduce that, for any element α ∈ SLf2(R) which is not a lifting from the identity, its conjugacy class is determined by three ] invariants: ` (α), the trace of α ∈ PSL2(R) and the direction type of α. The table above illustrates this. Compare it with the one presented in Theorem 4.6; notice that each n ≥ 1 represents a distinct conjugacy class, as well as each 0 < ϑ < π and λ > 1. We include the translation number of each conjugacy class as well, the calculation of which is a simple exercise. The central elements are highlighted in order to differentiate it from the non-central ones; notice that, when we include it in the table, the three invariants listed are no longer enough to distinguish all the conjugacy classes, for [Tn] and [Tn eu1] share the same description, as well as [T−n] and [T−n eu−1], for all n ≥ 1. Acknowledgements I would like to express my deep gratitude to Professor Shinichi Mochizuki and Professor Go Yamashita, my research supervisors, for having suggested me this topic, as well as for their patient guidance, encouragement, and useful critiques.

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RIMS, Kyoto University, 606-8502 Kyoto, Japan E-mail address: [email protected]