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Classification of the Conjugacy Classes of ˜ Sl2(R)

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Classification of the Conjugacy Classes of ˜ Sl2(R) 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 group of PSL2(R). For any non-central element α 2 SLf2(R), we show that its conjugacy class may be determined by three invariants: (i) Trace: the trace (valued in the set of positive real numbers R+) of its image α in PSL2(R); (ii) Direction type: the sign behavior of the induced self-homeomorphism 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 function 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 covering group 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 subgroup of Homeo+(S ), the group of orientation- preserving self-homeomorphisms 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 2 S1 ' R=Z, we show that the topological fundamental group 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 group extension 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 sequence by Z. 1 In Lemma 2.5, we show that this subgroup is actually the center 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 subgroups of Homeo] +(S ) that contain the subgroup 1 Z. More explicitly, we show that the conjugacy class of an element α 2 Homeo] +(S ) is determined uniquely by the following data: • an element ζ 2 Z, • the canonical lifting (as defined in Proposition 2.10) of a certain representative of 1 the conjugacy class of its image α 2 Homeo+(S ). This is stated in Corollary 2.11. The two main concepts used in the proof are the notions of CP-homomorphisms 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 kernel 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) := ja + dj, where [ c d ] = α 2 PSL2(R) is the image of α 2 SLf2(R) via the natural quotient map; 1 (ii) Direction type: Elements α 2 Homeo] +(S ) will be constructed in such a way that the function R 3 x 7! α(x) − x 2 R is invariant with respect to the translation R 3 x 7! x + 1 2 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 order to determine the behavior of α, it suffices to restrict our attention to x 2 [0; 1], which is compact. In particular, `(α) := supx2[0;1] jα(x)− xj is a well-defined real number. We then define the length-sharp function by taking `](α) := fb`(α)c; d`(α)eg ⊆ 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 integers. For x 2 R, we use the standard notations bxc := maxfk 2 Z j k ≤ xg and dxe := minfk 2 Z j k ≥ xg for the floor and ceiling functions, respectively. 0.1.1. Groups. For a group G, we denote its identity element by e 2 G and the center of G by Z(G). For a subset S ⊆ G, we denote the commutant (or centralizer) of S in G by ZG(S) := fg 2 G j gs = sg; 8s 2 Sg. Given g1; g2 2 G, we say that g1 −1 is conjugate to g2 if there exists h 2 G such that g1 = hg2h , in which case we write g1 ∼ g2. Conjugacy is an equivalence relation; we denote the conjugacy class of g 2 G by [g] := fh 2 G j g ∼ hg and the set of all conjugacy classes of G by C`(G). For any two elements g; h 2 G, we denote their commutator by [g; h] := ghg−1h−1. For elements gi 2 G, indexed by some set I 3 i, the subgroup generated by the gi's will be denoted by hgi j i 2 Ii. For N E G a normal subgroup, we denote the associated natural quotient map by G 3 g 7! g 2 G=N when no ambiguity is present. a b 0.1.2. Special linear group. An element ( c d ) 2 SL2(R) is a matrix with real entries satisfying ad − bc = 1, and we denote its image in PSL2(R) := SL2(R)={±Idg, where 1 0 a b −a −b Id := ( 0 1 ), by [ c d ] = −c −d 2 PSL2(R). Since the trace function on SL2(R) is a b given by tr(( c d )) = a + d, the function M 7! jtr(M)j 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 topological space 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) := ff 2 C(X; Y ) j f(K) ⊆ Ug, 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 component 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 category that admits finite products. For an object Y and a group object 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 morphism σ × pG×Y=Y : G × Y −! Y × Y; where pG×Y=Y : G × Y ! Y denotes the projection to the second factor, is an isomorphism. This is a natural generalization of the notion of principal G-bundles for smooth manifolds. 1. CP-extensions of groups 1.1. CP-homomorphisms. We start with a definition. Definition 1.1 (CP-homomorphism). A homomorphism of groups φ : G ! H is called commutant-preserving homomorphism (or a CP-homomorphism, for short) if 8x; y 2 G; φ(x)φ(y) = φ(y)φ(x) =) xy = yx: 4 CHRISTIAN TAFULA´ Remark 1.2.
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