Hyperbolic groups

Giang Le OSU,10/18/2010 Outline

• Quasi-isometry • Hyperbolic metric spaces • Hyperbolic groups • Isoperimetric inequalies • Dehn’s problems for groups • Rips complex and Rips theorem Quasi-isometry

(X, d) and (X’,d’) two metric spaces. f: X  X’ f is an isometry if d’(f(x),f(y)) = d(x,y) for all x,y. Note. If f is an isometry then f is connuous and injecve. If f is surjecve then X and X’ are called isometric.

Quasi-isometry is a weaker equivalence relaon. f is (C,k)-quasi-isometry if 1/C d(x,y) – k ≤ d’(f(x),f(y)) ≤ C d(x,y) + k

(X,d) and (X’,d’) are quasi-isometric if • There is a quasi-isometry f: X  X’ • Every point of X’ is in a bounded distance from the image of f Examples of quasi-isometric metrics spaces

• Every bounded is quasi-isometric to a point • (Z,d) and (R,d) • A finitely generated group G= with the word metric and its Cayley graph Γ=Γ(G,S) with the natural metric • If S and T are two finite generang sets of G then (G, ) and (G, ) are quasi-isometric Hyperbolic metric spaces Movaon: in hyperbolic plane H all triangles are thin. Want to extend the concept for a general metric space? What is a triangle in a metric space?

(X,d) is a metric space. A geodesic segment of length l in (X,d) is the image of an isometric embedding i: [0,l] -> X A (geodesic) triangle in X with verces x, y, z is the union of three geodesic segments, from x to y, y to z, and z to x respecvely. (X,d) is a geodesic metric space if there exist geodesic segments between all pairs of points.

A geodesic metric space is hyperbolic if there is a constant δ such that each edge of each triangle Δ in X is contained in the δ-neighborhood of the union of the other two sides of Δ

Examples: 1. Bounded metric spaces 2. Tree 3. Hyperbolic plane

Fact: (X,d) and (X,d’) are geodesic metric spaces, X and X’ are quasi-isometric. If X is hyperbolic then so is X’. Hyperbolic groups

G = a group. Γ = Γ(G,S) is the Cayley graph of G with the natural metric.

Def. G is called a if Γ is hyperbolic as a metric space. Note. The definion is independent from the representaon of G.

Examples 1. Finite groups 2. Finitely generated free groups 3. *Fundamental groups of compact Riemannian manifolds with strictly negave seconal curvature 4. *Groups that act cocompactly and properly disconnuously on a proper CAT(k) space with k<0

Non-examples 1. ZxZ 2. Every group which has ZxZ as its subgroup. Isoperimetric inequalies G = a group. S is finite. F(S) – with generator set S. w is a word in F(S), |w| is the length of w in F(S). If w = 1 in G then w = where and are word in F(S) Area(w)= the least value of n among all such expressions of w.

Group G is called hyperbolic if there is a constant k such that for every word w in F(S), w = 1 in G, we have an inequality Area(w) < k|w|

Examples: 1. Finite groups 2. Free groups Non-example 1. ZxZ

Fact 1. Two definions of hyperbolic groups are equivalent. Fact 2. Hyperbolic groups are finitely presented. Dehn’s problems for groups

Max Dehn (1878-1952), German mathemacian, a student of David Hilbert. (Dehn, 1911) The three fundamental problems for groups: 1. The identy problem 2. The conjugacy problem 3. The isomorphism problem

Note.(P. Novikov, 1955) There exists a finitely presented group G such that the word problem for G is undecidable

Hyperbolic groups are nice. Fact 1: The word problem is solvable in hyperbolic groups. Fact 2: The conjugacy problem is solvable for hyperbolic groups.

How about the isomorphism problem? (Sela, 1995) The isomorphism problem is solvable for torsion-free hyperbolic groups. ?(Dahmani, Guirardel, 2010, unpublished) the isomorphism problem for all hyperbolic groups. The Rips complex

(X,d) a metric space, α is a posive number. R(X,α) is a simplicial complex 1. The verces set is X 2. A finite set P of points of X forms a simplex if d(x,y) ≤ α for all x,y from P.

G = is a δ-hyperbolic group, d is the word metric on G. R(G,n) is the Rips complex for G, n is a posive integer.

Theorem (Rips). If n > 4δ + 2 then R(G,n) is contracble. Sketch of proof. Idea: show that every finite subcomplex is contracble to a point within R(G,n). Let K be any finite subcomplex of R(G,n), WLOG, 1 is contained in K.

Two cases: and Inducon on v.

Note: G acts freely disconnuously on R(G,n) and R(G,n)/G is compact. The end Thank you!