Connected components of representation spaces

Transcription

1 Connected components of representation spaces Kathryn Mann University of Chicago

2 Representation spaces Γ = finitely generated group G = topological group Hom(Γ, G) = space of homomorphisms Γ G. Natural topology as subset of G S S a generating set for Γ.

3 Hom(Γ, G ): key interpretations 1. Geometric structures to Riemann surfaces. Through a long development of the theory of hyperbolic affine spheres, culmim manifold, Γ = π (M), G Homeo(X ) nating with work of Labourie and Loftin, this space (where g is the genus of M) and, given Σ, the quadratic differentials form a complex vector space 1 form a space! C3g 3. Thus all the CP1 -structures homeomorphic to C6g 6. Furthermore, without even seeing one structure, one understands the whole moduli space globally as a cell of dimension 12g 12. naturally identifies with a holomorphic vector bundle over Teichmüller space whose fiber over a point #M$ is the space of holomorphic cubic differentials on M. An example of such a projectively symmetrical convex domain is depicted on the cover of the November 2002 issue of the Notices (See Figure 5). A (G, X )-structure on M is determined by ρ Hom(Γ, G ) (holonomy representation) e X and ρ-equivariant developing map, M dev Figure 3. A small deformation of this developing map maps M to a domain in CP1 that has fractal boundary. The corresponding representation is quasi-fuchsian, that is, topologically conjugate to the original Fuchsian representation. The developing map remains an embedding, and the holonomy representation embeds π1 (M) onto a discrete subgroup of PSL(2, C). In contrast Figure 4. As the deformation parameter [W. Goldman What is a projective structure? ] increases, the images of the fundamental octagons eventually meet and overlap

4 Hom(Γ, G): key interpretations 2. Space of Γ-actions M manifold, Γ = π 1 (M), G Homeo(X ) Hom(Γ, G) = space of Γ-actions on X G specifies regularity of action, e.g. G = Isom(X ), G = conf(x ), G = Diff r (X ). Regularity matters! Example: Hom(Z n, Diff 1 +(S 1 )) recently shown to be connected (A. Navas, 2013) Is Hom(Z n, Diff + (S 1 )) connected? Locally connected? (open)

5 Hom(Γ, G): key interpretations 3. Hom(Γ, G) = space of flat G-bundles M manifold, Γ = π 1 (M), G Homeo(X ) { flat X -bundles over M with structure group G } { representations π 1 (M) G } = Hom(Γ, G) bundle monodromy representation equivalent bundles conjugate representations

6 Connected components Connected components of Hom(Γ, G) correspond to deformation classes of structures actions bundles Basic question: classify components Example: G GL(n, C), Lie group Hom(Γ, G) is an affine variety finitely many components

7 Classical example: Hom(Γ g, PSL 2 (R)) Γ g := π 1 (Σ g ), PSL 2 (R) Homeo + (S 1 ) Theorem (Goldman, 1980) Components of Hom(Γ g, PSL 2 (R)) are completely distinguished by the Euler number, e(ρ). Milnor-Wood inequality (1958) ρ Hom(Γ g, PSL 2 (R)) (2g 2) e(ρ) 2g 2 Hom(Γ g, PSL 2 (R)) has 4g-3 components Two components of Hom(Γ g, PSL 2 (R)) are Teichmüller space = space of hyperbolic structures on Σ g. = set of discrete, injective representations Γ g PSL 2 (R) = components where e(ρ) is maximal/minimal Higher Teichmüller theory studies Hom(Γ g, G), G Lie group.

8 Hom(Γ g, G), G Homeo + (S 1 ) Known: G a Lie group. Hom(Γ g, S 1 ) is connected Hom(Γ g, PSL 2 (R)) has 4g 3 components, distinguished by e(ρ) Hom(Γ g, PSL (k) ) 1 Z/kZ PSL (k) PSL 2 (R) 1 Theorem (Goldman, 1980) Components of Hom(Γ g, PSL (k) ) are distinguished by e(ρ), unless k (2g 2) If k (2g 2), there are k 2g components where e(ρ) = ± 2g 2 k (the maximal/minimal values).

9 Flat circle bundles over surfaces G = Homeo + (S 1 ) What are the connected components of Hom(Γ g, G)? more representations (even up to conjugacy) but easier to form paths between representations Open Question Does Hom(Γ g, Homeo + (S 1 )) have finitely many components? Does Hom(Γ g, Diff + (S 1 )) have finitely many components?

10 Our results: Lower bound on number of components Theorem 1 (M-) For each divisor k ±1 of 2g 2, There are at least k 2g + 1 components of Hom(Γ g, Homeo + (S 1 )) where e(ρ) = 2g 2 k i.e. e(ρ) does not distinguish components and Hom(Γ 4, Homeo +(S 1 )) has 165 components... Moreover, two representations into PSL (k) that lie in different components of Hom(Γ g, PSL (k) ) cannot be connected by a path in Hom(Γ g, Homeo + (S 1 )).

11 A picture: ρ : Γ g PSL k with e(ρ) = 2g 2 k Start with ν : Γ g PSL 2 (R) with e(ν) = 2g 2. ν(b 1 ) ν(a 1 ) Lift to k-fold cover of S 1 for ρ : Γ g PSL k with e(ρ) = 2g 2 k

12 ρ(a 1 ) ν(a 1 )

13 Our results: Rigidity phenomena Theorem 2 (M-) Let ρ : Γ g PSL (k), e(ρ) = ±( 2g 2 k ). Then Connected component of ρ in Hom(Γ g, Homeo + (S 1 )) = Semiconjugacy class of ρ in Hom(Γ g, Homeo + (S 1 )) J. Bowden (2013): similar conclusion for Hom(Γ g, Diff + (S 1 )), fundamentally different techniques. Uses smoothness. Our key tool: rotation numbers Theorem (Rotation number rigidity; M-) Let ρ : Γ g PSL (k), e(ρ) = ±( 2g 2 k ), γ Γ g. Then rot(ρ(γ)) is constant under deformations of ρ in Hom(Γ g, Homeo + (S 1 )). rot(ρ(γ)) = rot(ρ t (γ))

14 Rotation numbers Definition (Poincaré) rot : Homeo + (S 1 ) R/Z. f rot(f ) := lim n (0) n n mod Z f Homeo Z (R) f Homeo +(S 1 ) continuous rot(f ) = p/q f has periodic point of period q rot(f m ) = m rot(f ) f Similarly, define rot : Homeo Z (R) R by rot( f ) := lim n (0) n n R Depends on lift f (not just f ). However, a commutator [f, g] Homeo + (S 1 ) has a distinguished lift [ f, g] so rot[f, g] makes sense.

15 rot is not a homomorphism! It is possible to have rot( f ) = rot( g) = 0 and rot( f g) = 1 Calegari-Walker (2011) give an algorithm to compute the maximum value of rot( f g) given rot( f ) and rot( g). [D. Calegari, A. Walker, Ziggurats and rotation numbers ] But... if f g = T n (translation by n), then rot( f ) + rot( g) = n

16 Proof ideas for rotation number rigidity (Theorem 3) Recall: Theorem 3 ρ : Γ g PSL (k), e(ρ) = ±( 2g 2 k ) rot(ρ(γ)) constant under deformations of ρ. Steps of proof: 1. The Euler number in terms of rot 2. Reduce to a question of local maximality of rot( f g) 3. Dynamics and the Calegari-Walker algorithm (4. Why rot and e(ρ) are key.)

17 The Euler number e(ρ) Classical definition is in terms of characteristic classes of circle bundles. e Z H 2 (Homeo +(S 1 ); Z). ρ (e Z ), [Γ g ] = e(ρ) Definition (Milnor) Γ g = a 1, b 1,...a g, b g [a 1, b 1 ][a 2, b 2 ]...[a g, b g ] ρ : Γ g Homeo + (S 1 ). e(ρ) := rot ([ ρ(a 1 ), ρ(b 1 )]...[ ρ(a g ), ρ(b g )]) e is continuous on Hom(Γ g, G) for any G Homeo +(S 1 ) [ρ(a 1 ), ρ(b 1 )]...[ρ(a g ), ρ(b g )] = id on S 1 [ ρ(a 1 ), ρ(b 1 )]...[ ρ(a g ), ρ(b g )] = T e(ρ)

18 Step 2. (A question of local maximality) ρ t path in Hom(Γ 2, Homeo + (S 1 )), ρ 0 as in Theorem. [ ρ t (a 1 ), ρ t (b 1 )] [ ρ t (a 2 ), ρ t (b 2 )] = T e(ρt) rot([ρ t (a 1 ), ρ t (b 1 )]) + rot([ρ t (a 2 ), ρ t (b 2 )]) e(ρ 0 ) If we show: rot([ρ t (a i ), ρ t (b i )]) has local max at t = 0, then we know rot([ρ t (a i ), ρ t (b i )]) is constant. From here, same kind of work shows that that rot(ρ t(a i )) and rot(ρ t(b i )) are both constant,...and also rot(ρ t(γ)) constant for any γ Γ.

19 Step 3. Dynamics and the Calegari-Walker algorithm f 0, g 0 Homeo + (S 1 ) f 0, g 0 Homeo Z (R). f t, g t deformations. Lift to paths f t, g t in Homeo Z (R) f t, g t f t, g t Study t rot( f t g t ). When is t = 0 a local maximum?

20 Step 3. Dynamics and the Calegari-Walker algorithm Toy example: f 0 g 0 Claim: rot(f 0 g 0 ) = 1/4 rot( f 0 g 0 ) = 1/4

21 Step 3. Dynamics and the Calegari-Walker algorithm dynamics at global maximum: at local maximum:

22 Why look at rot and e? rot and e essentially determine the dynamics of a representation. Theorem (Ghys) Γ any finitely generated group. ρ : Γ Homeo + (S 1 ) is determined (up to semiconjugacy) by the bounded Euler class ρ (e Z ) Hb 2 (Γ; Z). Theorem (Matsumoto) Γ any finitely generated group. ρ : Γ Homeo + (S 1 ) is determined (up to semiconjugacy) by the bounded Euler class ρ (e R ) Hb 2 (Γ; R) and the rotation numbers of a set of generators for Γ. For ρ : Γ Diff 2 +(S 1 ), determined up to conjugacy Ghys and Matsumoto s theorems let us use Rotation number rigidity to prove Theorems 1 and 2.

23 Work in progress Do other representations satisfy rigidity properties? Distinguish other components of Hom(Γ g, Homeo + (S 1 )). Example: is {ρ Hom(Γ g, Homeo + (S 1 )) : e(ρ) = 0} connected? Hom(Γ g, Homeo + (S 1 )) vs. Hom(Γ g, Diff + (S 1 )) What can analyzing rotation numbers tell us about components of Hom(Γ, Homeo + (S 1 )), for other Γ? (e.g. 3-manifold groups)