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Deforming Discrete Groups Into Lie Groups

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Deforming Discrete Groups Into Lie Groups Deforming discrete groups into Lie groups March 18, 2019 2 General motivation The framework of the course is the following: we start with a semisimple Lie group G. We will come back to the precise definition. For the moment, one can think of the following examples: • The group SL(n; R) n × n matrices of determinent 1, n • The group Isom(H ) of isometries of the n-dimensional hyperbolic space. Such a group can be seen as the transformation group of certain homoge- neous spaces, meaning that G acts transitively on some manifold X. A pair (G; X) is what Klein defines as “a geometry” in its famous Erlangen program [Kle72]. For instance: n n • SL(n; R) acts transitively on the space P(R ) of lines inR , n n • The group Isom(H ) obviously acts on H by isometries, but also on n n−1 @1H ' S by conformal transformations. On the other side, we consider a group Γ, preferably of finite type (i.e. admitting a finite generating set). Γ may for instance be the fundamental group of a compact manifold (possibly with boundary). We will be interested in representations (i.e. homomorphisms) from Γ to G, and in particular those representations for which the intrinsic geometry of Γ and that of G interact well. Such representations may not exist. Indeed, there are groups of finite type for which every linear representation is trivial ! This groups won’t be of much interest for us here. We will often start with a group of which we know at least one “geometric” representation. To make things more precise, we can assume the existence of a representation i :Γ ! G which is discrete and faithful. In other worlds, we assume that Γ can be realized as a discrete subgroup of G. We will for instance consider the following groups Γ: • the free group Fn in n generators, • the fundamental group Γg of a closed surface of genus g ≥ 2, 3 4 • the fundamental group of a closed hyperbolic manifold, • the group SL(n; Z) ⊂ SL(n; R), • the Coxeter groups. One the framework is settled, our course will be motivated by the follow- ing questions: Question 0.0.1. Let Xbe a G-homogeneous space. What can be said of the action of i(Γ) on X? Since i(Γ) is discrete, it acts properly discontinuously on G by left mul- tiplication, and more generally on every homogeneous space G=K where K is a compact subgroup of G. On the other hand, an infinite subgroup Γ n n of Isom(H ) certainly does not act properly on @1H , which is compact. n One can however hope to find a nice “limit set” ΛΓ ⊂ @1H , on which the dynamics is well-understood, and outside of which the action is properly discontinuous. Question 0.0.2. Does the representation i :Γ ! G admit non-trivial de- formations ? One can always deform i by conjugating it with elements of G. Such deformations are “trivial” in the sens that the induced actions of Γ on some G-homogeneous space are conjugate via a transformation of G. This question will lead us to introduce the character variety, which is the quotient of the space of representations of Γ into G under the conjugation action of G. If i is isolated in the character variety, we call i rigid . During the second half of the 20th century, many groundbreaking works have contributed to show how many discrete subgroups of Lie groups are rigid. A striking example is that every representation of SL(3; Z) is rigid (as a consequence of Margulis’s superrigidity theorem).We will state those theorems, mostly to exclude those groups of our study. We will then focus on the discrete groups that can actually be deformed. In particular: • representations of the free group Fn easily deform, since a representa- tion of Fn into G is given by any n-tuple of elements in G, • surface groups also have rich character varieties, which carry a sym- plectic structure invariant under the action of the mapping class group of the surface, • in some cases, we can deform the fundamental group of a closed hy- perbolic manifold of dimension n ≥ 3 into a Lie group that contains n stricly Isom(H ). Finally, when we know how to deform a discrete and faithful representa- tion i :Γ ! G we can ask the following: 5 Question 0.0.3. Which geometric and dynamical properties of i(Γ) are pre- served under deformation? In particular, are deformations of i still discrete and faithful? Is their action on G-homogeneous spaces still topologically con- jugate to that of i ? We will introduce the notion of Anosov representation of a discrete group Γ, for which the answer to this question is affirmative in some way. This notion was introduced 15 years ago as a unified setting in which to study: n • convex cocompact subgroups of Isom(H ), • divisible convex sets, • Globally hyperbolic Cauchy compact anti-de Sitter manifolds. A beautiful example: Fuchsian and quasi-Fuchsian representations Let S be a closed oriented surface of genus g ≥ 2 and Γ its fundamental 2 group. Let us consider first representations of Γ into the group Isom+(H ) 2 of orientation preserving isometries of H . This group is isomorphic to PSL(2; R) = SL(2; R)={±Id2g. Among these representations, the ones that are discrete and faithful are called Fuchsian. Let j be a Fuchsian representation. Then j(Γ) acts freely and properly 2 2 discontinuously on H , so that Σ = j(Γ)nH is a smooth hyperbolic surface. Besides, j induces an isomorphism between Γ and π1(Σ). It follows that Σ diffeomorphic to S. In other words, j defines a homotopy class of hyperbolic metrics on S. How many such metrics are there? Poincaré’s uniformization theorem provides an answer: as many as isotopy classes of complex structures on S. Theorem 0.0.4 (Poincaré). Let Σ be a Riemann surface of genus at least 2. Then there exists a unique conformal metric on Σ of constant curvature −1. (A conformal metric is a metric that writes αdzdz in local coordinates.) Putting all this together, we obtain a homeomorphism between the Te- ichmüller space T (Σ) of isotopy classes of complex structures and the space RepFuchs(Γ; PSL(2; R)) of Fuchsian representations modulo conjugation. To complete the picture, one would want to characterize which rep- resentations of Γ into PSL(2; R) are Fuchsian. For this, one introduces the Euler class eu(ρ) of a representation ρ (which describes the topol- ogy of the PSL(2; R)-principal bundle associated to ρ). The Euler class eu : Rep(Γ; PSL(2; R)) ! Z is continuous, hence constant on connected components. The following theorem gathers the properties of the Euler class and its relation with the topology of the character variety Rep(Γ; PSL(2; R)): Theorem 0.0.5. • The Euler class takes values between 2 − 2g and 2g − 2 (Milnor–Wood, [Mil58]), 6 • For every k 2 f2−2g; : : : ; 2g−2g, eu−1(k) is non-empty and connected (Goldman,[Gol88]) • For every k 2 f1;:::; 2g − 2g, eu−1(k) is homeomorphic to a vector bundle of rank 4g − 4 + 2k over the symmetric power Sym2g−2−k(Σ) (Hitchin, [Hit87]), • A representation ρ is Fuchsian if and only if jeu(ρ)j2g − 2 (Goldman, [Gol80]). In particular, discrete and faithful representations form a connected com- ponent in Rep(Γ; PSL(2; R)). What happens now if we deform a Fuchsian representation into PSL(2; C) ' 3 Isom+(H )? For small deformations, we obtain quasi-Fuchsian representa- tions. These act properly discontinuously and cocompactly on the comple- 3 ment of a Jordan curve in @1H . The quotient of this domain of disconti- nuity gives two Riemann surfaces homeomorphic to S. This defines a map from the space of quasi-Fuchsian representations (modulo conjugation) to T (S) × T (S). A remarkable theorem of Ahlfors–Bers shows that this map is bijective! We thus obtain a complete description of the set of “nice” repre- sentations of Γ into PSL(2; C). However, quasi-Fuchsian representations do not form a connected component in Rep(Γ; PSL(2; C)), and there is a whole research field devoted to understanding limits of quasi-Fuchsian. We will come back to this beautiful example, which the theory of Anos representations intend to generalize to Lie groups of higher rank. Chapter 1 Semisimple Lie groups and their symmetric spaces This first chapter briefly introduces the first steps of the structure theory of semisimple Lie groups. 1.1 A Reminder of Lie theory Let us first recall without proofs the basics of Lie theory. We encourage the reader who is not familiar with this material to consider the proofs as exercises. 1.1.1 Lie algebras Definition 1.1.1. A Lie algebra g over a field K is a vector space over K endowed with a bilinear operation [·; ·]: g × g ! g ; called the Lie bracket, which satisfies the following properties: • antisymmetry: [u; v] = −[v; u], • Jacobi identity: [u; [v; w]] + [v; [w; u]] = [[w; u]; v]. Example 1.1.2. Let A be an algebra over K. Then the Lie bracket [u; v] = uv − vu endows A with a Lie algebra structure. For instance. Example 1.1.3. Let M be a smooth manifold. Then the space of smooth vector fields on M, with the Lie bracket of vector fields, is a Lie algebra. 7 8 CHAPTER 1. LIE GROUPS AND SYMMETRIC SPACES Remark 1.1.4.
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