Implications of hyperbolic geometry to operator K-theory of arithmetic groups

Transcription

1 Implications of hyperbolic geometry to operator K-theory of arithmetic groups Weizmann Institute of Science - Department of Mathematics LMS EPSRC Durham Symposium Geometry and Arithmetic of Lattices, July 6th, 2011

2 Implications of hyperbolic geometry to operator K-theory of arithmetic groups This talk will be about The Mathematical topics connected by the Baum/Connes assembly map An interesting example: the Bianchi groups Motivations for studying Bianchi groups The system of representation rings of their finite subgroups The equivariant K-homology of the Bianchi groups Torsion subcomplexes of the Bianchi groups

3 The (analytical) assembly map Paul Frank Baum Alain Connes µ i : K G i (EG) K i (C r (G)), i N {0}

4 The Bianchi groups Definition For m a positive square-free integer, let O m denote the ring of algebraic integers in the imaginary quadratic field extension Q[ m] of the rational numbers. The Bianchi groups are the projective special linear groups Γ := PSL 2 (O m ). Motivations Group theory Hyperbolic geometry Knot theory Automorphic forms Baum/Connes conjecture Algebraic K-theory Heat kernels Quantized orbifold cohomology

5 The modular tree for PSL 2 (Z) Underlying picture by Robert Fricke for Felix Klein s lecture notes, 1892

6 The PSL 2 (Z)-equivariant retraction

7 A fundamental domain for Γ = PSL 2 ( Z[ 37 ] ) PSL 2 (Z) PSL 2 (R) H 2 R PSL 2 (O m ) PSL 2 (C) H 3 R

8 Complex representation rings of the cell stabilisers Character tables. Let j = e 2πi 3. Z/2 1 g ρ ρ A 4 1 (12)(34) (123) (132) χ χ j j 2 χ j 2 j χ Frobenius reciprocity: (φ τ ) G = (φ τ) H

9 The Bredon chain complex 0 σ Γ \X (2) R C (Γ σ ) Ψ 2 σ Γ \X (1) R C (Γ σ ) Ψ 1 σ Γ \X (0) R C (Γ σ ) 0

10 Equivariant K-homology Theorem (R.) Let Γ := PSL 2 (O m ). Then, for O m principal, the equivariant K-homology of Γ has isomorphy types m = 1 m = 2 m = 3 m = 7 m = 11 m {19, 43, 67, 163} K Γ 0 (EΓ) Z 6 Z 5 Z/2 Z 5 Z/2 Z 3 Z 4 Z/2 Z β 2 Z 3 Z/2 K Γ 1 (EΓ) Z Z 3 0 Z 3 Z 3 Z Z β 1, where the Betti numbers are m β β

11 Extracting the torsion subcomplexes

12 The non-euclidean principal ideal domain cases

13 Geometry of the torsion elements Theorem (R.) For any vertex v H, there is a natural bijection between the Γ rotation axes passing through it and the non-trivial cyclic subgroups of its stabiliser. Corollary (R.) For any vertex v H, the action of its stabiliser on the set of Γ rotation axes passing through it, restricted from the action of Γ on H, is given by conjugation of its non-trivial cyclic subgroups.

14 Geometry of the torsion elements Theorem (R.) Let v be vertex in hyperbolic 3-space. Then the number n of orbits of subdivided edges adjacent to v, with stabiliser in Γ isomorphic to Z/lZ, is given as follows for l = 2 and l = 3. Isomorphy type of Γ v {1} Z/2Z Z/3Z D 2 S 3 A 4 n for l = n for l =

15 Fritz Grunewald ( ) P l (t) := q = vcd(γ)+1 dim Fl H q (Γ; Z/l) t q.

16 Group homology Theorem (R.) The l primary part of the integral homology of PSL 2 (O m ) depends in degrees greater than 2 (the virtual cohomological dimension) only on the homeomorphism type of the l torsion subcomplex.

17 The results in homological 3 torsion Let Pm(t) 3 := ( ) dim F3 H q (PSL 2 OQ[ m ] ; Z/3)t q. q=3 m specifying the Bianchi group 3 torsion subcomplex, homeomorphism type P 3 m(t) 2, 5, 6, 10, 11, 15, 22, 29, 34, 35, 46, 51, 58, 87, 95, 115, 123, 155, 159, 187, 191, 235, 267 2t 3 t 1 7, 19, 37, 43, 67, 139, 151, 163 t 3 (t 2 t+2) (t 1)(t 2 +1) 13, 91, 403, t 3 (t 2 t+2) (t 1)(t 2 +1) 39 2t 3 t 1 + t3 (t 2 t+2) (t 1)(t 2 +1)

18 Thanks Thanks a lot for your attention!