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The Combinatorial Structure of Cocompact Discrete Hyperbolic Groups*

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The Combinatorial Structure of Cocompact Discrete Hyperbolic Groups* JAMES W. CANNON THE COMBINATORIAL STRUCTURE OF COCOMPACT DISCRETE HYPERBOLIC GROUPS* This paper is dedicated to Larry and Helen 0. INTRODUCTION Combinatorial group theory began with Dehn's study [7]-[9] of the fundamental group of the closed 2-dimensional manifold admitting a hyperbolic structure. At first Dehn freely used arguments from hyperbolic geometry, but rapidly he and others moved in the direction of results which could be stated and proved in purely combinatorial and algebraic terms. Thurston (see [20], [21]) has recently shown that large classes of groups of interest to topologists, while not obviously amenable to attack by standard methods of combinatorial group theory, nevertheless are discrete hyperbolic groups. His result suggests the value of a return to geometric considerations in combinatorial group theory. We shall show that Dehn's principal combinatorial theorems-in parti- cular his solutions to the world and conjugacy problems for hyperbolic surface groups-have simple geometric reinterpretations, and that these solutions, as reinterpreted, are true for all cocompact, discrete hyperbolic groups. We shall also show that the global combinatorial structure of such groups is particularly simple in the sense that their Cayley group graphs (Dehn Gruppenbilder)(see [4]-[9]) have descriptions by linear recursion. We view this latter result as indicating a promising generalization of small cancellation theory (see [14, Chap. V]) where small cancellation hypotheses may fail locally but, in some sense, hold globally. The result also indicates that cocompact, discrete hyperbolic groups can be understood globally in the same sense that the integers Z can be under-stood: feeling, as we do, that we understand the simple linear recursion n ~ n + 1 in Z, we extend our local picture of Z recursively in our mind's eye toward infinity. One obtains a global picture of the arbitrary cocompact, discrete hyperbolic group G in the same way: first, one discovers the local picture of G, then the recursive structure of G by means of which copies of the local structure are integrated. The results of this paper need to be complemented by specific com- *This research was supported in part by a research grant of the National Science Foundation and in part by a Romnes faculty fellowship at the University of Wisconsin, Madison. Geometriae Dedicata 16 (1984) 123-148. 0046-5755/84/0162-0123503.90. © 1984 by D. Reidel Publishing Company. 124 JAMES W. CANNON putations. Dehn made some such computations in his papers [7]-[9], others appear in our previous papers [1], [2], and still more will appear in joint work with Thurston and Grayson. All that we need to assume about hyperbolic geometry is collected in an appendix. Such an appendix would have been unnecessary in Dehn's day. Thurston proved an early version of the recursion theorem. We learned most of our hyperbolic geometry from his notes on 3-manifolds, and we appreciate his notes, ideas, and encouragement. After fixing the setting, we shall make a series of hyperbolic estimates in Sections 1, 2, 3, and 4. All the main results follow quickly from these estimates. The main theorems are proved in Sections 5, 6, and 7. Some unsolved pro- blems are summarized in Section 8. Settino Good background references are [12] and the Appendix. We assume throughout that H denotes hyperbolic space of some fixed dimension with d n its hyperbolic invariant metric (see Appendix). We also fix for con- sideration a group G of hyperbolic isometries acting discretely on H (if K c H is compact, then {g ¢ G [gK ¢~ K # ~ } is finite) with compact orbit space H/G. We choose a finite generating set C for G (see [16]) and let F = (G, C, E) denote the group graph of G with respect to the generating set C of G (see [2], [3], [5], [7]). We let d r denote the word metric on F (each edge has length 1). We emphasize that the metric is to be defined on the entire graph and not just on the vertices so that the graph becomes a connected metric space. We fix a point 0 of H such that G acts freely on 0. We define a map ~l :F ~ H as follows: on vertices, 9~G, q,(g) = g(o) and on edges, (g, c, ffc), g E G, c ~ C, ~,(0, c, g.c) = [g(0), g.c(0)],, where [a, b]n denotes the hyperbolic geodesic segment joining points a and b of H. If f e G, then f(dp(g)) = f "g(O) = (a(f .g) and f dp(g, c, g'c) -- fig(O), g'c(O)]. = [f'g(O), f'g'c(O)] x = dp(f'g, c, f'g'c) DISCRETE HYPERBOLIC GROUPS 125 so that 4~ F ,H (f ") fl F ,H commutes. That is, 4) is a G-equivariant map that is injective on vertices. Since G acts discretely on H, it follows that ~b is proper (i.e. 4)- 1 (compact) = compact). The graph F and its image ~b(F) in H are objects of surpassing beauty. Generally, F and ~b(F) may be identified with the graph dual to a tessellation of H by a fundamental polyhedron associated with G. Fricke initiated drawings of a number of such tessellations. These drawings, or slight modi- fications, may be found for example in [11], [13], or [15]. We point out to the reader two distance-related objects which we find useful. The first comes into play in Theorem 2: if L is a subset of hyperbolic space H and W>0, then Nbd(L, W)= {peH:dH( p, L)<~ W}. The second object first appears in Section 3: if L is a hyperbolic line, I c L, and W > 0, then the W-corridor about I along L is the set W(I; L) = {zeH: dn(z, L) <~ W and the orthogonal projection of z into L lies in I}. l. QUASI-COMPARABILITY OF METRICS THEOREM 1. There exist positive numbers k and k' such that (l.l) dr(x, y) ~ k.dn(gp(x), (a(y)) and (1.2) dn(c~(x), 4)(Y)) ~< k'dr(x, Y) for all x, yc F satisfying either dr(x, y) >1 k' or d.(q~(x), ~)(y)) >~ k'. Proof (See Figure 1.) Since H/G is compact and ~b is G-equivariant, • ' L_._ • Ix) ~(yl Figurel. dr~<k'd n. 126 JAMES W. CANNON there is a number ~ such that each point of H is within 0t of an element of ~b(F). Since H/G is compact, there exists a compact set K of H such that any two points of H at H-distance ~< 2~ + 1 lie in some G-translate of K. Since ~b is proper, the set ~b-1K is compact and has a finite dr-diameter 6. Since q~ is G-equivariant, it follows that dr(P, q) <<. 6 whenever du(ck(p), dp(q) ) <<. 2~ + 1. Now let x, yeF. Let [~b(x), ~b(y)] n denote the hyperbolic geodesic segment joining ~b(x) and ~b(y) in H, and let dp(x) = Po, Pl ..... pj --= ~b(y) denote points of [~b(x), ~b(y)]n at most one unit apart such that J j - 1 d.tp,_l, P,) = i=1 Let q~(x) = qo, ql, '", qj, "-, qj = ~b(y) be points within 0t of Po, Pl ..... Pj, respectively, such that qiE ~b(F). Let x = qo,I q~t ..... qjt = y denote ~b preimages of qo, ql .... , qj in F. Then J (1.3) dr(x, y) <<. ~. d r (q;_,, q;)<<, j" 6 <<. tS-dH(~b(x), ~b(y)) + 6. i=l Now if da(~b(x), ~b(y))>t 1, it follows from (1.3) that dr(x,y ) <<. 26 dn(~b(x), ~b(y)). If dr(x, y) >~ 26, then it follows from (1.3) that dr1(~b(x), ~b(y)) 1> 1 so that again dF(x, y) <<. 26 dR(q~(x), dp(y) ). These observations show that (1.3) implies (1) for k = 26 and k' = max{l, 26}. (See Figure 2.) Since ~b is G-equivariant and F/G is compact, it follows that there is an e > 0 such that the ~b-image of each edge of F has H-dia- meter ~< e. Let x, yeF and let P denote a path of minimal length joining x to y in F. Let x = x o, x 1 .... , x~_ 1, xj = y denote edges or segments of edges in F whose union is P such thatj - 2 ~< ~{= l dr(xi_ 1, xi) = dr(x, Y)" Then J (1.4) dn(O(x), ~b(y)) ~< ~ da(dp(x i_ 1), ~b(x,)) ~< j'e ~< e'dr(x, y) + 2E. i=1 Then (1.2) follows from (1.4) as (1.1) followed from (1.3). $(x) = $(x 0) Figure 2. d u <~ k'd r. DISCRETE HYPERBOLIC GROUPS 127 2. QUASI-GEODESICS IN F Hyperbolic geodesics (written H-geodesics) are unique. Minimal paths in F, which we shall call F-geodesics, are not necessarily unique. Theorem 1 shows that F-geodesics are what we shall call (k, k')-quasi-geodesics. DEFINITION 1. Suppose that K and K' are positive numbers. A path P in F is called a (K, K')-quasigeodesic if every subpath Q of P having F-length IQI ~> K' has ~b-image in H whose endpoints are at hyperbolic distance >f (1/K)'I Q I" We consider only subpaths of edge paths. We show here that (K, K')-quasi-geodesics have ~b-images in H which roughly follow H-geodesics.
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