Introduction to Hyperbolic Groups

Question 1

Discuss motivation for hyperbolic groups

Answer 1

1. Fundamental groups of closed negatively curved manifolds
2. Combinatorial group theory - small cancelation groups

pg 1-2

Question 2

Discuss appropriate category for hyperbolic groups/geometric group theory

Answer 2

OBJECTS:
(X,d) - geodesic metric spaces
Examples:
1. Complete Riemannian Manifolds
2. Connected graphs - edge length 1
in particular Cayley graphs of f.g. groups

MORPHISMS:
Def. (1) f(X,d) –> (Y,p) is a (k, eps)-quasi-isometric embedding if for all x1, x2 in X

1/k d(x1, x2) - eps <= p(f(x1), f(x2)) <= kd(x1, x2) + eps

(2) f is a (k, eps, C)-quasi-isometry if
(a) f is a (k, eps) QI-embedding
(b) f is C-coursely surjective i.e. Y <= N_c(f(X))

equivalent to C close to identity

pg 2-4

Question 3

Show Cay(G, S) and Cay(G, S’) are quasi-isometric where S and S’ are different (symmetric) generating sets

Answer 3

Write one set of generators in terms of others - only changes length by a linear factor…pg 4-5

Question 4

Milnor-Schwarz lemma? Proof?

Answer 4

Special case: M complete Riemannian manifold, G f.g. group acting geometrically on M, then (M,g) quasi-isometric to G.

pg 5 - 7

Question 5

Define: hyperbolic geodesic metric space, hyperbolic group

Answer 5

delta-hyperbolic - thin triangles

group hyperbolic if Cay(G,S) hyperbolic for some generating set S

pg 8

Question 6

Discuss distance between a path and a geodesic segment in a hyperbolic pace

Answer 6

Bridson 1.6 419-420

Question 7

Prove: Hyperbolicity is a QI invariant

Answer 7

Fix proof with Chris or BH
pg 8-9

Question 8

What is Morse Lemma? Proof?

Answer 8

Quasi-geodesics track geodesics in hyperbolic metric spaces

pg 10, 13

Question 9

Show Morse Lemma => Hyperbolicity is a QI invaraint

Answer 9

pg 11-12

Question 10

Discuss notions equivalent to delta hyperbolicity

Answer 10

Slim = Thin = Insize

pg 14-16

Question 11

Discuss divergence function

Answer 11

pg 16-17