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Hyperbolic Groups Hyperbolic groups Giang Le OSU,10/18/2010 Outline • Quasi-isometry • Hyperbolic metric spaces • Hyperbolic groups • Isoperimetric inequaliBes • 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 con2nuous and injecve. If f is surjecve then X and X’ are called isometric. Quasi-isometry is a weaker equivalence relaBon. 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=<S|R> with the word metric and its Cayley graph Γ=Γ(G,S) with the natural metric • If S and T are two finite generaBng sets oF G then (G, ) and (G, ) are quasi-isometric Hyperbolic metric spaces MoBvaBon: 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 verBces x, y, z is the union oF three geodesic segments, From x to y, y to z, and z to x respecBvely. (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 = <S|R> 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 defini2on is independent from the representa2on of G. Examples 1. Finite groups 2. Finitely generated free groups 3. *Fundamental groups oF compact Riemannian maniFolds with strictly negaBve secBonal curvature 4. *Groups that act cocompactly and properly disconBnuously on a proper CAT(k) space with k<0 Non-examples 1. ZxZ 2. Every group which has ZxZ as its subgroup. Isoperimetric inequaliBes G = <S|R> 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 definiBons oF hyperbolic groups are equivalent. Fact 2. Hyperbolic groups are finitely presented. Dehn’s problems for groups Max Dehn (1878-1952), German mathemaBcian, a student oF David Hilbert. (Dehn, 1911) The three Fundamental problems For groups: 1. The idenBty 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 posiBve number. R(X,α) is a simplicial complex 1. The verBces 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 = <S|R> is a δ-hyperbolic group, d is the word metric on G. R(G,n) is the Rips complex For G, n is a posiBve integer. Theorem (Rips). If n > 4δ + 2 then R(G,n) is contracBble. Sketch of proof. Idea: show that every finite subcomplex is contracBble 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 InducBon on v. Note: G acts freely discon2nuously on R(G,n) and R(G,n)/G is compact. The end Thank you! .
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