Hyperbolic groups
Giang Le OSU,10/18/2010 Outline
• Quasi-isometry • Hyperbolic metric spaces • Hyperbolic groups • Isoperimetric inequali es • 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 con nuous and injec ve. If f is surjec ve then X and X’ are called isometric.
Quasi-isometry is a weaker equivalence rela on. 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 metric space 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 genera ng sets of G then (G, ) and (G, ) are quasi-isometric Hyperbolic metric spaces Mo va on: 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 ver ces x, y, z is the union of three geodesic segments, from x to y, y to z, and z to x respec vely. (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 hyperbolic group if Γ is hyperbolic as a metric space. Note. The defini on is independent from the representa on of G.
Examples 1. Finite groups 2. Finitely generated free groups 3. *Fundamental groups of compact Riemannian manifolds with strictly nega ve sec onal curvature 4. *Groups that act cocompactly and properly discon nuously on a proper CAT(k) space with k<0
Non-examples 1. ZxZ 2. Every group which has ZxZ as its subgroup. Isoperimetric inequali es G = a group. S is finite. F(S) – free group 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 defini ons of hyperbolic groups are equivalent. Fact 2. Hyperbolic groups are finitely presented. Dehn’s problems for groups
Max Dehn (1878-1952), German mathema cian, a student of David Hilbert. (Dehn, 1911) The three fundamental problems for groups: 1. The iden ty 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 posi ve number. R(X,α) is a simplicial complex 1. The ver ces 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 posi ve integer.
Theorem (Rips). If n > 4δ + 2 then R(G,n) is contrac ble. Sketch of proof. Idea: show that every finite subcomplex is contrac ble 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 Induc on on v.
Note: G acts freely discon nuously on R(G,n) and R(G,n)/G is compact. The end Thank you!