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~ 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) = $(x 0) Figure 2. d u <~ k'd r. DISCRETE HYPERBOLIC GROUPS 127 2. QUASI-GEODESICS IN F Hyperbolic geodesics (written H-geodesics) are unique. Minimal paths in F, which we shall call F-geodesics, are not necessarily unique. Theorem 1 shows that F-geodesics are what we shall call (k, k')-quasi-geodesics. DEFINITION 1. Suppose that K and K' are positive numbers. A path P in F is called a (K, K')-quasigeodesic if every subpath Q of P having F-length IQI ~> K' has ~b-image in H whose endpoints are at hyperbolic distance >f (1/K)'I Q I" We consider only subpaths of edge paths. We show here that (K, K')-quasi-geodesics have ~b-images in H which roughly follow H-geodesics. THEOREM 2. Given K and K' > O, there is a positive constant W such that if x and y are any two points of F, if L is a hyperbolic line through q~(x) and th(y), if Nbd(L, W) = {p~H :dn( p, L) <<. W}, and if P is any (K, K')-quasi- 9eodesic from x to y in F, then ok(P) c Nbd(L, W). Proof. The idea for the proof comes from Mostow's rigidity theorem (see [17] and [20, Chap. V].) The key estimate is the fact that hyperbolic orthogonal projection p : H ~ L descreases 1-dimensional measure in H by at least the factor sech s, where s is the hyperbolic distance to L (see Proposi- tion A.7, Appendix). As a consequence it is much faster to move from qS(x) to qb(y)even at slow speed along L than to move from qS(x) to ~b(y)at higher speed along a path which leaves a relatively small neighborhood of L. The precise estimates are as follows (see Figure 3): ~ /¢(bl Figure 3. ]~b(Q)I is bounded. 128 JAMES W. CANNON We first fix positive numbers e and s such that (2.1) diameter ($(e)) < e and 1 (2.2) sech s < K-~e' where e varies over all edges of F. We then take x, y, L, and P as in the statement of the theorem and set N s = Nbd(L, s). We let Q denote a subpath of P having q~-image of maximal hyperbolic length [~(Q)l such that ~#(Q)c H- Int N~. We assume that IQ[ ~> K'. We let a and b denote the end points of Q. Then inequality (2.1), Definition 1, and Proposition A.7 show that (I/Ke) Iq~(Q)I <~ (1/K) IQ[ <~ du(~p(a), (b) ) <~ 2s + (sech s) ldp(Q)[. Hence (2.2) implies that ]~(Q)[ ~< ~--~- sech s .2s. Consequently, we conclude that if W = s + max {~K',(K---~-sechs)-~2s}, then q~(P) c Nbd(L, W). 3. QUASI-GEODESIC PROGRESSION IN CORRIDORS (See Figure 4.) Let x and y denote distinct points of H and L = L(x, y) the line in H through x and y. We may parametrize L by an isometry $ : R --, L which takes 0 to x and dn(x, y) to y. This parameterization allows us to con- sider intervals such as [a, b], - oo < a < b ~< oo, and expressions such as x + N as lying in L. Given a line L in H, a subset I c L, a positive number W, and orthogonal projection p : H -~ L, we define the W-corridor about I along L to be the set W(I;L) = {zeH[d(z, L) <<, W and p(z)~l}. Theorems 1 and 2 allow us to make the following estimates. THEOREM 3. Suppose that K, K', and W are as in Theorem 2 and that M isapositive number. Then there is a positive number N satisfying thefollowinff conditions: DISCRETE HYPERBOLIC GROUPS 129 Y~ -t~ QD Figure 4. The W-corridor about L(x, y). If x and y are points F, if L is a hyperbolic line through dp(x) and ~(y) in H, if P is a (K, K')-quasi-geodesic from x to y in F, and if p, p'~P, then (3.1) p( dp(p) ) = p( dp(p') ) implies dp(p, p') < N; (3.2) de(x, y) > N implies dn(c~(x), ~b(y)) > 2M; (3.3) de(x, p) > N implies dp(p)e W((dp(x) + M, oo); L); (3.4) dp(x, p) > N and dl,(p, y) > N together imply that L c~ Nbd(q~(p), W)= (~b(x) + M, ~b(y)- M)c~ Nbd(~b(p), W) ¢ ~b, where de(a, b), a, b~P, denotes the length of the subpath of P from a to b, and where Nbd(~b(p), W) = {zEH :dH(qS(p), z)~< W}. REMARK. If any of conditions (3.1)-(3.4) is satisfied for one value of N, it is satisfied for all larger values of N. Hence, in applications of Theorem 3 we need only require in (3.2), (3.3), and (3.4) that dp(x, y) >1 N. Proof of (3.1). By Theorem 2, dH(~b(p), ~b(p') ~< 2W. If the subpath Q of P from p to p' has length 1> K', then the definition of (K, K')-quasi-geodesic implies that dH(~b(p),~b(p'))/> (1/K).dp(p, p'). Hence de(p, p') <<,max (r', K.2W). Proof of (3.2). We take N = max(K', K- 2M). Since dl,(x, y) > N >/K', the definition of (K, K')-quasi-geodesic implies that dn(c~(x), c~(y)) >t (l/K). dp(x, y) > 2M. Proof of(3.3). If p($(p)) ~< tp(x) in L, then there is a point p"eP such that de(x, p") >>-de(x, p) and such that p(~b(p")) = p($(x) ) = $(x). But then dp(x, p) <<, de(x, p") < N', where N' satisfies (3.1). If ~b(x) ~< p(t~(p)) ~< $(x) + M in L, 130 JAMES W. CANNON then either de(x, p) < K' or de(x, p) <<. K.dn(ck(x), c~(p)) <<. K.(M + W). In any case, if we set N > max {N', K', K(M + W)}, then c~(p)eW((c~(x) + M, 0o); L). Proof of (3.4). We apply (3.2) and (3.3) to find an N such that de(x, p) > N and de(p, y) > N imply together that ~b(x)+ M + W < p((a(p)) < ~(y) - M - W. By Theorem 2, Nbd(~b(p), W)n L =fi ~b. But by the inequality just established Nbd(qS(p), W) c~ L c (q~(x) + M, ~b(y) - M). 4. LOCAL QUASI-GEODESICS The main theorems of this paper are consequences of the fact that, with certain restrictions, local quasi-geodesics are global quasi-geodesics (Theorem 4) and hence, approximately follow H-geodesics (Theorem 2): THEOREM 4. Given K, K' > O, there exist K", L, and L' > 0 having the following property. If P is a path in F and if each subpath of P of length <. K" is a (K, K')-quasi-geodesic, then P is an (L, E)-quasi-geodesic. In addition to Theorem 2 and 3 we require the following two hyperbolic angle and distance estimates, Lemmas 1 and 2. LEEMA 1. Given W > 0 and M > O, there is an angle ~, 0 < ~ < ~, having the following property. Let x and z denote points at a distance >1 2M from each other on a hyperbolic line L in H. Let y denote a third point of H at distance <. W from L whose orthogonal projection y' on L lies in the subinterval Ix + M, z - M] of L (see Section 3). Then the angle ~' between the geodesic segments [x, Y]H and [y, z]n at y is at least ~t. Proof. (See Figure 5.) After a hyperbolic isometry we may assume that L is the vertical ray from the origin 0 in ~", that the W-neighborhood of L in H is the Euclidean cone C with vertex 0 and axis L (see Proposition A.7, Appendix), and that x lies below z on L. Then y is restricted to the circular sector in C bounded by the Euclidean circles C 1 and C 2 with center 0 through x + M and z - M, respectively. The hyperbolic isometry S :H --} H, S:u-}r.u, with r < 1 chosen so that r'yeC 1, preserves the angle ~t'. The angle S(~t') may be reduced by fixing S(y), sliding S(z) down to the point x + 2M, and sliding S(x) upward to x. Thus the minimum admissible value of ~' occurs with z = z + 2M, yeCl. Let ~ denote that minimum, and the proof of Lemma l is complete. LEMMA 2. Given angles fl and 7, 0 < fl, ~ <. ~, and a number R > 0, there is a length D > 0 having the following property. Suppose [a, b]n is a hyper- bolic geodesic in H of hyperbolic length D'>~ D joining a, bell such that DISCRETE HYPERBOLIC GROUPS 131 / Figure 5. The angle estimate. [a, b]n makes an anole 7' >>"7 with the upward directed vertical ray at a. Then (4.1) [a, b]n makes an anole fl' <~ fl with the vertical at b, and (4.2) log (d(a)/d(b))=R'>>-R, where d(x) denotes the Euclidean distance from x to dH = R n- i. Proof. For a and b on the same vertical line, it follows that 7' = n, ff = 0, and R' = D' (see Proposition A.5, Appendix), so that it suffices to take D = R in that case. (See Figure 6.) For a and b on a dH-orthogonal semicircle C of Euclidean radius r centered at y~dH = R n- 1, we introduce Euclidean polar coordinates in the vertical Euclidean 2-plane through C so that a = (r, 7') and b = (r, n - if), ~,' < n - ft. Using the formula of Proposition A.6 (Appendix) we calculate the following: (4.3) D' = log tan((rt - if)/2) - log tan(7'/2 ) ~< log tan((n - fl')/2) - log tan(7/2 ) Choosing D so large that (4.4) D + log tan 0,/2) i> log tan ((n - fl)/2) we find that rc - fl' >/n - fl so that fl' ~< ft. This proves formula (4.1) in this case. Another calculation, using Proposition A.6 (Appendix), yields 132 JAMES W. CANNON , a 0 '"\\\~"~\C I/ ~/ ', y r Figure 6. The angle-distance estimate. (4.5, D' -log(sin-- sin~-,,))+log( y' 1 -cos(Ir-fl')~l~,J 1 - cos - =log(d(a)/d(b))+ log -l ~os~ ] ~ (4.6) D - log 1 - cos ~ we fined that (4.5) establishes formula (4.2). Hence Lemma 2 is true for D satisfying D > R, (4.4), and (4.6). Proof of Theorem4. (See Figure 7.) The positive numbers K and K' determine W > 0 via Theorem 2. We choose positive numbers M and R arbitrarily. The numbers K, K', W, and M determine N > 0 via Theorem 3. The numbers W and M determine an angle ~ 0 < ~ < 7r, via Lemma 1. Taking y = ~/2 and fl = ~/2, the angles fl, ~, and the number R determine D > 0 via Lemma 2. By (3.2) we any increase N so that dp(x, y) I> N implies dH($(x), $(y)) >I D. We let e denote the maximal length for $(e), e an edge of F. We then choose K", L, and/~ subject to the conditions DISCRETE HYPERBOLIC GROUPS 133 2 I r~3 Figure 7. Quasi-geodesic. (4.7) K" = 2N (4.8) LR - N > 0 (4.9) L' = [1 + (L.e.N + N)/(LR - N)].N We now prove that any path P in F which is K"-locally a (K, K')-quasi- geodesic is, in fact, globally an (L, L')-quasi-geodesic. We assume P has length I PI > L'. We choose points x 0 ..... Xk+ 1 in P such that (4.10) de(xi_l,xi)=N (i<,k) (4.11) de(Xk, Xk + 1) <~ N k+l (4.12) IP] = Z de(x,-vx,)=k'N +de(Xk, Xk+l )~(k + 1)N. i=1 It follows from (4.9) and (4.12) that (4.13) k >I (L'e" N + N)/(LR - N). By (4.7), each of the paths P[x i, xi+ 2] is a (K, K')-quasi-geodesic, i = 0, ..., k-1. Let y~ denote the point 4~(xi), i = 0 ..... k + 1, and let [y~_ 1, Yi]H denote the H-geodesic joining Yi-1 and Yi in H. We may assume, after an isometry of H, that Yo lies vertically above Yl- Let 0£, 0t2 .... denote the angles 134 JAMES W. CANNON between [Yo, Yl]n, [Yl, Y2]H.... at Yl, Y2 ..... respectively. Let fll = 0, f12, -" denote the angles between [Yo, Yl]n, [Yl, Y2]n .... and the upward vertical ray at Yl, Y2 ..... respectively. Let 71 = ~tl, 72, --. denote the angles between [Yl, Y2]rx, [Y2, Y3]n .... and the upward vertical ray at y~, Y2 ..... respectively. By (3.2) and (3.4), Lemma 1 applies to the three points Yi-1, Yi, Y/+ 1, for all i~< k- 1. Hence ~t~/> ~t for all i ~< k- 1. It is true for 71 = ~1 ~> 0t = 27, and, we assume it to be true inductively for 7~-1, that 7~-1 ~> 7. Then, by Lemma 2, fl/~< fl = ~/2. Hence 7//> ~i - fli ~> ~ - ~/2 = ct/2 = 7. We conclude that 7//> 7 for all i ~< k - 1. Hence, since dn(y ~_ 1, Y/) >~ D by choice of N, Lemma 2 implies that log(d(y/_l)/d(y/))>~R for i= 1..... k, where d(y) is the Euclidean height of y above dH. We conclude that (4.14) dn(Yo, .~k + 1) ) dlt(Yo, Yk) -- dH(Yk, Yk + 1) i> log (d(Yo)/d(Yk) ) - e" N >~k'R-e.N. But (4.12) implies that (4.15) (1/L)'de(xo, Xk+ 1) <~ (1/L)'(k + 1)N. Since k'R - e.N >1 (1/L)(k + 1)N for k >>-(L'e'N + N)/(LR - N) (which it is by (4.13), it follows from (4.8), (4.9), (4.13), (4.14) and (4.15) that (4.16) dn(Yo, Yk+ 1) >~ (1/L)dl"(xo, xk+ 1), which completes the proof of Theorem 4. 5. DEHN'S SOLUTION TO THE WORD PROBLEM See references [7]-[9]. Let K denote the group with presentation (x~ 1, y~l ..... x~l,y~ 1 :R =[xl,yl] ... [Xk, Yk] = 1),k> 1. Oehn proved that any nonempty word W in the generators x~ 1, y~ 1.... , x k± 1,Yk . ± 1 representing the identity element in K can be shortened by one of two processes: (1) delete a trivial word xx-1 from W; or (2) for some cyclic permutation R' - A. B- 1 of R, with length A > length B, replace A in W by B. (A word W which cannot be so shortened is called Dehn-reduced.) One way of paraphrasing the process is as follows: Nonempty words W representing the trivial element of K can be shortend by means of relators xx- 1 of length 2 and relators R' of length 4k. Returning to our hyperbolic group G with graph F, we prove the following analogous result. DISCRETE HYPERBOLIC GROUPS 135 THEOREM 5. There is an integer R such that nonempty words W in the generating set C of G representing the identity element of G can be shortened by means of relators of G having length ~ R. (A word which cannot be so shortened is called R-reduced.) Proof. By Theorem 1 there exist positive numbers K and K' such that F-geodesics are (K, K')-quasi-geodesics. The numbers K, K' determine numbers K", L, and L' via Theorem 4. Choose R > 2 max {K", L' }. Suppose that some word W of length/> R/2 cannot be shortened by means of some relator of length ~< R. This means precisely that every subword of length ~< R/2 is represented by a path in F which is a F-geodesic. Since K "<<, R/2, it follows that W is K'-locally a (K, K')-quasi-geodesic. Consequently, W is represented by a path P of length > L' which is an (L, L')-quasi-geodesic. Hence the end points x o, x I of P have ~b-image in H at distance dH(~b(x0), ~b(x1 )) >i (1/L)dp(xo, x I ) >1 L'/L > 0. Thus x o ~ x 1 and W does not represent the identity element of G. 6. DEHN'S SOLUTION TO THE CONJUGACY PROBLEM (See Figure 8.) Our generalization, Theorem 6, of Dehn's solution 1-9] to the conjugacy problem for surface groups requires for its proof that we study bi-infinite words U~= ( .... u_ 1 , u o, u 1 .... ) in the generators C of G. Usually such words arise as bi-infinite powers ... U" U" U .... Of course, if one begins at any vertex vo(U ~) of F, there is an infinite path P+ beginning at vo(U ~) and labelled u0, u 1 .... and an infinite path p-1 begin- ning at vo(U) and labelled u_l,u_2,-1-1 ..... Then the path P=P P+ re- presents U ~ and is based, as we shall say, at oo (U®). We may label the remain- ing vertices of P as .... v_l(U~),vo(U~),v~(U®),.., so that vi(U~°).u~ , ~.-gt,t,(PI)---,I I ~(P_) a Figure 8. A bi-infinite word and its axis. 136 JAMES W. CANNON = vi+l(U~)(-0o Any finite word U in the generators C of G gives rise to a bi-infinite word U ~° .... U-U-U .... Similarly, each finite path P in F gives rise to a bi- infinite path poo based at the initial vertex of P; if U labels P, then U ® labels P®. An axis for P~ is also called an axis for P. LEMMA 1. Let L, L' be positive numbers. Then any bi-infinite path P® which is an (L, L')-quasi-ffeodesic has an axis L(a, b). Furthermore, dp(P) lies in the W-neighborhood of L(a, b), where W > 0 is the positive number deter- mined by L and L' via Theorem 3. Proof Let .... v 1, Vo, Vl, ... denote the vertices of P. By definition of (L, L')-quasi-geodesic, de(v o, v i) -~ 0o implies dH(d?(Vo), dp(vi)) ~ 0o. Hence the sequence ~b(v 1), ~b(v-2) .... clusters at at least one point a~Hu R n- 1 u {0o} and (p(vI ), ~b(v2)~ ... clusters at at least one point b. Ifa = b, then for ~b(v_i) and ~b(vi) very near to a= b,L(~p(v_i), cp(vi)) has hyperbolic W- neighborhood N very near a = b in the Euclidean sense. In particular, we may choose i so large that ~b(Vo)¢N; a contradiction to Theorem 2. Hence a =~ b. A similar argument shows that a and b are the only cluster points so that lim~_._oo~b(vi)= a and limi_.o0~p(vi)= b exist and are distinct. If N i is the closed hyperbolic W-neighborhood of the hyperbolic line L(~p(v_i), ~(v~)) and N is the closed hyperbolic W-neighborhood of the hyperbolic line L(a, b), then the N i converge to N in the Euclidean sense. Hence, since, for fixed x, dp(x)~N i provided that x~.P~Ev_i , vi] by Theorem 2, it follows that ~b(P~) c N. It remains to show that p.d? :poo ~ L(a, b) is an (M, M')- quasi-isometry for some M and M'. This follows easily from the fact that ~b is a quasi-isometry by Theorem 1, P ---, F is a quasi-isometry onto its image, and piN is a proper map with point preimages of constant finite diameter. DISCRETE HYPERBOLIC GROUPS 137 LEMMA 2. Suppose that P is a finite path in F with label T, initial point Xo, and terminal point x I . Let U denote a finite word, and let Qo and Q1 denote paths in F, each labelled by U, Qi beginning at xi, i = O, 1. Let Q denote a finite path with label V from the terminal point Y0 °fQo to the terminal point Yl °f Ql " Then if P has an axis L(a, b), the line L(a, b) is also an axis for Q. Proof. If .... x 1, x0, xl, "'" are points of P~, such that x i" T = xi+ 1 (-oo Proof. Let K, K', K", L, L', and R all be as in the first paragraph of the proof of Theorem 5. Let U, V, and X 0 denote words, U and V cyclically R-reduced, such that X o t UX ° and V represent the same element of G. Let P denote the path based at the identity vertex 0 of G with label U, and let Q denote the path based at 0" X 0 and labelled V. Case 1. Suppose one of the paths poo and QO~ has an axis L(a, b). Then both have L(a, b) as axis by Lemma 2. Either [P] ~< R, in which case q~(poo) lies in the N = cosh- 1(e" R/e') + e" R neighborhood of L(a, b) as established in the proof of Lemma 3, or I P[ >>,R >t 2K". In the latter case each subpath of length ~< K" is a F-geodesic, hence a (K, K')-quasi-geodesic, hence P~ is an (L, L')-quasi-geodesic, hence p~o lies in the W-neighborhood of L(a, b) by Lemma 1, W determined by L and L' only. We find that in Case 1, P® and QO~ share an axis and lie within the (M = max{N, W} )-neighborhood of that axis. There is thus by Theorem 1 a path P' leading from a vertex of p~o to a vertex ofQ ~° with [P'[ <~max{k'+2k.M} +2. IfgeG is the trans- formation of H which by left multiplication shifts P® and QO~ one P-block and one Q-block to the right, if U' is the label of the path in P® cut off by P' and gP', V' is the label of the path in QOO cut off by P' and gP', and if X is the label of P', then U', V', and X are the words promised by the conclusion of Theorem 6, with IxI max{k' + 2k.max {N, W} } + 2. Case 2. Suppose neither P nor Q has an axis. Then I uI, I vI ~ R. Thus there are only finitely many choices for U and V. Thus any two such U and V which are in any way conjugate are conjugate by a word of length ~< R' for some R' independent of U and V. We may take this R'>>. max{k'+ 2kmax{N, W} } + 2. Then the conclusion of Theorem 6 is satisfied in both Case 1 and Case 2. 7. LINEAR RECURSION IN HYPERBOLIC GROUPS We define a partial order ~< on F by defining x ~ y if and only if there is a F-geodesic from the identity vertex 0 of F to y which passes through x. It is convenient for our purposes to have ~< defined on all of V and not just on the vertices of F. If e = (v, c, v.c) is an edge of F, then either dr(0, v) + 1 = d(O, v'c), in which case v < v.c and the edge e is said to be norm-in- creasing, or dr(0, v)- 1 = d(0, o'c), in which case v > v'c and the edge e is said to be norm-decreasing, or dr(0, v)= d(0, v'c), in which case e is said to half-increase and half-decrease norm (if x is in the first half of such an edge, then x > v; ifx is in the last half, x and v are not comparable in (F, ~<)). We define intervals [x, y], [-x, oo), [0, x] in F, relative to ~<, in the usual way; for example, [x, oo)= {yeF:y/> x}. We call [x, ~) the cone at x. DISCRETE HYPERBOLIC GROUPS 139 Two vertices x and y are said to have the same cone type in F if the graph automorphism taking x to y given by left multiplication by yx- t in G takes [x, ~) isomorphically onto [y, oo) and preserves the (increasing, decreasing) order type of the edges (or half-edges) in [x, oo). Two vertices x and y are said to have the same N-type in F, N > 0, if the graph automorphism T : F --} F taking x to y has the property that (7.1) dr(x', 0) - dr(x, 0) = dr(Tx', 0) - dr(Y, 0) for each x' e F satisfying dr(x', x) ~ N. LEMMA 1. There is an N > 0 such that if x and y are vertices of F having the same N-type, then x and y have the same cone type in F. Proof. By Theorem 1 there exist positive numbers K and K' such that F-geodesics are (K, K')-quasi-geodesics. By Theorem 4 there are positive numbers K", L, and L' such that K"-locally (K, K')-quasi-geodesics are (L, L')-quasi-geodesics. The pairs (K, K') and (L, L') each determine a positive number W via Theorem 2; we take W so large that it works for each pair. The triples (K, K', W) and (L, L', W), together with an arbitrary positive number M, each determine a positive number N via Theorem 3; we take N so large that it works for each triple. Finally we make N larger than max{K", 2WK, K', L'} + 1. We shall show that N satisfies the requirements of Lemma 1 (see Figure 9). Suppose, therefore, that x and y are vertices of F having the same N-type. Let X and Y denote the N-neighborhoods of x and y in F, respectively. Let T: F-} F denote the F-automorphism taking x to y. Let z~[x, oo) and 0 at ~ X x 1 t .Z I Figure 9. The cone type of a vertex. 140 JAMES W. CANNON let P denote a path from x to z that is strictly norm-increasing. We must show that P'= T(P) is also strictly norm-increasing. We let Q and Q' denote strictly norm-increasing paths from 0 to x and form 0 to y = T(x), res- pectively. Any subpath of Q'u P' of length ~< K" either lies in Q' or in (Q'u P')n Y or in T(P)= P' since N >t K". Hence Q' uP' is K"-locally a (K, K')-quasi-geodesic, hence also an (L, L')-quasi-geodesic. Let R' denote a F-geodesic joining the end points of Q' u P'. Then R' is also a (K, K')- quasi-geodesic. We claim that R' intersects the N-neighborhood Y of y. If not, then the distance in Q' w P' from y to the end points 0 and z' = T(z) of Q' uP' is bigger than N. Hence, formula (3.4) implies that the W- neighborhood of ~b(y) intersects the H-geodesic between qb(0) and ~b(z') between ~b(0)+ M and ~b(z')- M. Formula (3.4) also implies that ~b(R') lies in the W-neigborhood of the geodesic line through ~b(0) and ~b(z'). It follows that some point of ~b(R') lies within 2W of ~b(y). Hence dn(dp(y), q~(R')) ~< 2W. Thus Theorem 1 implies that dr(Y, R') <~ max{K', 2W'K} < N- 1. Hence R' has a vertex v' in Y. Let ~', ff denote the subpaths of R' from 0 to v' and form v' to z', respectively. Let ~ denote a norm-increasing path from 0 to v = T-l(v') and fl = T-1(if). Let R = ~ufl. We are now prepared to deduce the fact that the length IQ'I +lP'l of Q'uP' equals I~'1 ÷ I~'1 = IR'I, which of course was chosen to be dr'(0, z'). The relevant facts are (7.2) I~1-1~'1 =IQI-IQ'I (7.3) IBI=I~'I (7.4) IPI = IP'l (7.5) IQl+lel=dr(O,z)<~l~l+l~ I (7.6) I~"l+l~'l=lR'l=dr(O,z')~lQ ' +IP'I Statement (7.2) follows from (7.1) since x and y have the same N-type, (7.3) and (7.4) since T preserves lengths, (7.5) and (7.6) from the definition of F-geodesic. Using in turn (7.6), (7.2), (7.4), (7.5), and (7.3), we find I~'1 +IB'I ~<[O'l+lP'l --I~'l-I~l+lQl+lel ~l~'l-I~l+l~l+l~l =l~'l+l~'l Hence I~'l + I~'l = I Q'l ÷ IP'l as desired. This completes the proof of Lemma 1. DISCRETE HYPERBOLIC GROUPS 141 COROLLARY 2. There exist only finitely many equivalence classes of vertices in F if two having the same cone type are considered to be equivalent. Proof. It is clear that there exist only finitely many N-types for fixed N. Therefore Lemma 1 implies the desired result. REMARK. Corollary 2 contains the heart of our assertion made in the introduction that the graph F can be described by a linear recursion. We have made that assertion in spite of the fact that we do not quite know what the assertion means except on the basis of experience with many examples over the last two and a half years. Roughly it should mean that F can be put together in layers out of finitely many types of building blocks and that, by examining what has happened in the previous layer, we know which building blocks to use at the next layer. It means that building a new layer never causes the collapse of previously constructed layers except within themselves, and then only very near the layer we are working in. Careful examples appear in [2] and [3]. The connection with Corollary 2 is that the sets [x, ~), of which there are only finitely many types, are determined by finite initial segments, and these segments may be used as the building blocks mentioned. The types of the immediate successors to x are determined by the type of x, hence by the building block associated with x. The new blocks at a given layer unfortunately must overlap, but the overlapping is completely determined by certain local rules true in any Cayley group graph. These rules account for the limited amount of surface collapsing. In practice, we seem to be able to visualize very completely the entire process. Since our personal experience does not seem adequate to suggest the best formulation, we restrict ourselves to the following final result. DEFINITION. Ifa i denotes the number of vertices v off such that dr(0, v) = i, then f(x) = E~°=o a~xf is called the growth function of F. THEOREM 7. The growth function f (x) off is a rational analytic function. REMARK. Particular growth functions of Cayley graphs are discussed in [1] and [3]. Proof. Without changing f(x), we may delete if necessary enough of the generators C so that no color represents the identity element of G and no two colors represent the same element of G. Then F has no loops or double edges. Suppose x and y are vertices of F joined by a norm-increasing edge e=(x,c,y) so that y~[x, ~). Then [y, ~)c Ix, ~) so that the cone type t(y) of y is completely determined by c and the cone type t(x) of x. In parti- cular, the number n(t(x), t(y)) of vertices y' that, like y, are terminal end 142 JAMES W. CANNON points of a norm-increasing edge from x with type t(y') equal to t(y) depends only on the type t(x) of x. Similarly, the number ngt(y)) of vertices x' that, like x, are initial points of a norm-increasing edge from x' to y depends only on the type of y. Indeed, if k is the number of elements of C u C- t and k(y) is the number of norm- increasing or half-increasing edges from y, then k(y) depends only on the type t(y) of y, and m(t(y) ) = k - k(y). Let t o, tt, ..., t s denote the cone types of vertices in F, with t o denoting the type of the identity vertex 0. Then the equivalence class t o consists of the singleton {0}, since 0 is the only vertex y with m(t(y)) = 0. Let a, i denote the number of vertices y such that dr(Y, 0) = n and t(y) = t i. Define n u = n(ti, t) and m~ = re(t). Then we can easily see that the following linear recursive relationships are valid: {10 f°rn=O (7.7) an° = otherwise (7.8) a0j = 0 forj :# 0 N (7.9) a~ = (l/m) ~ noa ._1,i for n > 0 and j 4= 0 i=O Setting f.(x)j = E~= o a~FJ, we find that N (7.10) f(x) = ~ f j(x) j=O (7.11) fo(X) = 1 N (7.12) f j(x)--(1/m).x. ~. niJ~(x)" i=0 It is a routine problem in linear algebra to solve equations (7.10), (7.11), and (7.12) for fo, ft ..... fN, and f. The result is a rational function as claimed. 8. OPEN QUESTIONS Some important problems remain to be solved. (1) Carry out the analogous study for Euclidean groups and Kleinian groups with cusps. (An easier problem is to carry out the study for funda- mental groups of compact, closed mainfolds of negative curvature and for groups cocompact in the convex hulls of their limit sets (e.g. finitely generated free groups); this easier problem is routine and involves only slightly more sophisticated estimates.) DISCRETE HYPERBOLIC GROUPS 143 (2) Formalize the notion that a Cayley graph can be described by linear recursion, and devise efficient algorithms for working out that recursion for many examples. Calculate explicit growth functions for many examples. Derive the analytic properties of those growth functions and relate them to the analytic and measure theoretic properties of the hyperbolic groups and their limit sets. (See [3], [12], [18], [19].) (3) Connect this study will small cancellation theory and work out the consequences. (See [14].) (4) Use similar geometric ideas to study the automorphism groups of hyperbolic groups with the aim of finding a generalization of Thurston's theorem on canonical representations for elements of the mapping class groups of surfaces. (See [21].) (5) Find the combinatorial properties of G and F that mirror the other major idea in the proof of Mostow's rigidity theorem, namely ergodicity of the action of G on the limit set of F. (The first major idea of Mostow's proof forms the heart of this paper: see Section 2.) (See [17]-[20].) (6) Characterize (cocompact) discrete hyperbolic groups combinatorially. A start is indicated in Floyd's thesis [10]. Thurston's work on hyperbolic structures on anannular Haken manifolds should supply major hints [22]. (7) Apply explicit growth functions to the study of orbital counting functions [12, p. 108]. APPENDIX ON HYPERBOLIC GEOMETRY Hyperbolic space is the smooth submanifold H={x=(xl, ...,x)e~": x > 0} of Euclidean space R" equipped with the hyperbolic Riemannian metric ds 2 = x~- 2 dx 2, where dx 2 = dx2~ + -.. + dx 2 is the standard Euclidean metric. A diffeomorphism f :H ~ H is called a hyperbolic isometry if it preserves the hyperbolic metric: f*(ds2)=f~2.(df2+ ... +df2)=ds 2. Hyperbolic geometry is the theory of the invariants of the group of hyperbolic isometrics of H. Straightforward calculation shows that the following diffeomorphisms of H are hyperbolic isometrics; in fact they generate the group of all hyperbolic isometrics. (A.1) Euclidean isometrics of H:T(x)=(t(Xl, ...,x 1),x), where t: ~"- 1 --* R"- a is a Euclidean isometry. (A.2) Euclidean similarities of H : S(x) = rx, r > O. (A.3) Euclidean inversion of H in the Euclidean unit sphere S"-1 of I~":I(x) = (x 2 + ... + X ,)2 - 1 "X. A hyperbolic k-plane in H is the intersection with H of a Euclidean k-plane or k-sphere which meets OH --- R"- ~ orthogonally. 144 JAMES W. CANNON PROPOSITION A. 1. lf P is a hyperbolic k-plane in H, k < n, and iff : H ~ H is a hyperbolic isometry, then f (p) is a hyperbolic k-plane. Proof. A hyperbolic k-plane P is the intersection of (n - k) hyperbolic (n- 1)-planes. Thus it suffices to consider the case k = n- 1. Since A.1 is clearly true for f of type (A.1) or (A.2), we may assume f is the inversion 1 (see (A.3)). Any hyperbolic (n - 1)-plane has the form P = {X eH :a(X. X) + B. X + c = 0}, where (.) denotes the Euclidean inner product, a and c are scalars, B is a vector in dH, and X is a variable vector. The Euclidean (n - 1)- plane case is the case a = 0, B :~ 0. The Euclidean (n - 1)-sphere case is the case a --/: O, B.B - 4ac > 0. One easily sees that I(P) = {XeH:c(X.X) + B. X + a = 0}, and that either c = 0 and B ~ 0 or c --p 0 and B" B - 4ac > 0. This observation completes the proof. PROPOSITION A.2. Each hyperbolic k-plane P in H is totally geodesic in the sense that any two points of P are joined by a unique #eodesic in H, and that geodesic lies in P. Proof. It suffices to show the existence of a retraction p :H ~ P which strictly reduces 1-dimensional hyperbolic measure ds at points of H- P. Proposition A. 1 makes it clear that, after a Euclidean isometry and inversion of H, we may assume that P = {Pl ..... Pk- 1' O, .... O, p,)eH}. We introduce polar coordinates in H relative to P: x e H is assigned coordinates (p, q, r, ~b), where p(x) = (xl ..... x~_ 1), S (Xk .... , X,_ 1) if (Xk,..., X n- 1) = O, q(x) = x((X2 dr.. • " + X n_1)-1/2"(x~2 ..... x 1) otherwise, r(x) = (x~ +... + x2W 2 r(x).cos ~(x) 0 if(xk, ..., X _ 1) = 0, and ( X2k "~ "'" + X n-2 1 )1/2 otherwise (0 < ~b ~< n/2). If follows easily that x 1 = Pl, ".., Xk-x = P~-I, Xk = r'COS q~'qk ..... X.-1 --- r'cos q~-q._ 1, and x.--r.sin ~b. Therefore, for points of H- P, calculation yields the formula ds 2 -- dp 2 d- r 2 cos 2 t~ dq 2 q- r2d~b 2 -t- dr 2 r 2 sin 2 ~b with dp 2 = dp~ + "" + d Pk-2 1 and dq 2 = d qk2 + "'" "}" dq,_2 1 . (The calculation DISCRETE HYPERBOLIC GROUPS 145 uses the observation that q2 + ... + q2_1 = 1 so that qk dqk +"" + qn-1 dqn- 1 = 0.) The formula p(p, q, r, ~b) = (19, 0, r, n/2) defines a retraction p : H ~ P along hyperbolic (n- k)-planes orthogonal to P. If dt 2 denotes the hyperbolic Riemannian metric in P, we have (A.4) dt(p, 0, r, n/2) 2 - dp2 rE+ dr2 ~< sin 2 ~b-ds(p, q, r, ~b)2. Since sin2 ~b < 1 for (19,q, r, ~b)~H- P, p is the desired measure reducing retraction. COROLLARY A.3. Hyperbolic lines (hyperbolic 1-planes) are the only hyperbolic geodesics. COROLLARY A.4. If xeH- P, then p(x) is the unique point of P nearest to x. Proof. The hyperbolic line L through x and p(x) is orthogonal to P. There is a hyperbolic (n - 1)-plane P' containing P which is also orthogonal to L. Euclidean reflection R of H through P' is a hyperbolic isometry which leaves L invariant. Hence L contains the unique geodesic between x and R(x). If 7 were any other path from x to P' of length as short as the geodesic from x to p(x), then 7 w R(7 ) would be as short as the geodesic from x to R(x); a contradiction to Corollary A.3. The hyperbolic distance between two points z 0 and z I of H is defined by the formula (A.5) dH(Zo, zl) = ~ ds, O where ?o is the unique geodesic from z o to z 1 . PROPOSITION A.5. If z o and z 1 are points of the same vertical line L in H with d 1 = (Euclidean distance from z 1 to OH)> d o = (Euclidean distance from z o to all), then d~ dt (A.6) d(z o, z 1 ) = I -- = log (d 1/d o ). dao t Proof. By Corollary A.3, the vertical line L contains the unique geodesic 7o from z 0 to zl. This path 7o inserted in (A.5) yields the desired formula. PROPOSITION A.6. If z o and z 1 are points lying on a dH-orthogonal semi- circle C in H, and if C is given Euclidean polar coordinates (r, ~b), r = Euclidean 146 JAMES W. CANNON radius of C, 0 < qb < 7r, z o = (r, dpo), z 1 = (r, q~l ), qbo < qbl, then 1 cos 0 (A.7) dH(Zo, Zl)=jc, osinq ~ ~log i+cos~- 1 l+c°sq~0/ = log tan (~bI/2) - log tan (t~o/2) Proof. By Corollary A.3, C contains the geodesic Y0 from z o to z 1 . Insertion in (A.5) yields formula (A.7). PROPOSITION A.7. Let p :H ~ P denote the retraction defined in the proof of Proposition A.2. Then the factor sin ~ by which p reduces 1-dimensional measure ds (as calculated in (A.4)) is precisely sech (dH (x, P) ). Proof. By Corollary A.4, dH(X, P) = dH(x, p(x)). By Proposition A.6, c-ff-s-s But cosh log 1 - cos PROPOSITION A.8. If P is the vertical line in H throu#h the origin in R" and e > O, then the set of points of H at hyperbolic distance < e from P forms an open riyht circular cone with vertex 0 and axis P. Proof. It suffices to examine the orbit of a single point x~H- P under those Euclidean isometries of H (see (A.1)) which fix P and under the Euclidean similarities of H which leave P invariant (see (A.2)). Alternatively, the proof of Proposition A.7 shows dH(X, P) depends only on the polar angle ~b relative to P. PROPOSITION A.9. The set of points having constant hyperbolic distance from a point p of H forms a Euclidean (n - 1)-sphere. We do not include a proof. The easiest involves another model for hyper- bolic space, namely the disk model D. In that model, with p the center of the disk, the result is obvious. But the hyperbolic transformations taking D to H take Euclidean spheres to spheres. PROPOSITION A.10. Euclidean and hyperbolic angle measures coincide. Proof. If r/and ~ are tangent vectors to H at x~H, then the hyperbolic inner product is given by 0/, ~)n = (~/1 ~1 + "'" + t/,~,)" 1/X2. Hyperbolic angle measure is defined by cos(~bn) = (t/, ¢)n/(~l, ~)1/2(~, ~)1/2. After cancella- tion of the x,, this equals the Euclidean measure cos (~bE) = r/. ~/] 11¢1. PROPOSITION A.11. Hyperbolic angles, arc lengths, and volumes are DISCRETE HYPERBOLIC GROUPS 147 given by the following formulas invariant under hyperbolic isometr y : (A,8) cos( ) (Euclidean measure) I (A.9) length y = f~ ds # (A.10) volume A = f dxl "'" dx. X n J, 4 n Proof. Taking Proposition A.10 into account, formulas (A.8), (A.9), and (A.10) arise from ds via standard formulas of Riemannian geometry. Thus they are preserved by transformations preserving ds. NOTES ADDED NOVEMBER, 1983: (1) The analysis of Sections 1-5 can be applied to geometrically finite groups. First, however, we must augment the Cayley graph so as to fill in combinatorial horoballs. The details will appear elsewhere. (2) The finitely many shortening relators whose existence is asserted by Theorem 5 have been calculated explicitly in only a few cases. Dehn found them for closed surface groups. W. Magnus purports to do the same thing for triangle groups in Hyperbolic Tesselations (Academic Press). Unfortunate- ly his result is incorrect as stated, as examples exhibited by Tatsuoka show. Tatsuoka has shown how to correct the Magnus result and has made a number of other specific calculations. (3) Max Benson (preprint) has shown that all Euclidean groups have only rational growth functions. This suggests that these groups satisfy the stronger condition of having only finitely many vertex cone types. This has been verified for all finite generating sets of free abelian groups (Cannon, unpublished). Matthew Grayson (Princeton University Thesis, 1983) has extended the list of groups with rational growth functions. (4) The theorems on finiteness of vertex ,:one types are a natural general- ization of the normal form theorems for free products with amalgamation and HNN extensions (Cannon, unpublished). REFERENCES 1. Bourbaki, N. : Groupes et algebres de Lie, Chs 4, 5 et 6, Hermann, Paris (1968). 2. Cannon, J. W. : 'Colored Graphs' (Preprint). 3. Cannon, J. W. : 'The Growth of the Closed Surface Groups and the Compact Hyperbolic Coxeter Groups' (Preprint). 148 JAMES W. CANNON 4. Cayley, A. : 'On the Theory of Groups', Proc. London Math. Soc. 9 (1978), 126-133. 5. Cayley, A. : 'The Theory of Groups: Graphical Presentations', Amer. J. Math. I (1878), 174-176. 6. Cayley, A. : 'On the Theory of Groups', Amer. J. Math. 11 (1889), 139-157. 7. Dehn, M. : '~ber die Topologie des dreidimensionalen Raumes', Math. Ann. 69 (1910), 137-168. 8. Dehn, M. : 'Ober unendliche diskontinuierliche Gruppen', Math. Ann. 71 (1912), 116-144. 9. Dehn, M. : 'Transformation der Kurve auf zweiseitigen Flgtchen', Math. Ann. 72 (1912), 413-420. 10. Floyd, W. J. : 'Group Completions and Kleinian Groups', Ph.D. Dissertation, Princeton Univ., 1978. 11. Fricke, R. and Klein, F. : Vorlesungen i~ber die Theorie der automorphen Funktionen, Vol. 1 (1897) and Vol. 2 (1912), Teubner, Leipzig. (Reprinted by Johnson Reprint, New York, 1965.) 12. Harvey, W. J. : Discrete Groups and Automorphic Functions, Academic Press, London, 1977. 13. Lehner, J.: Discontinuous Groups and Automorphic Functions, Amer. Math. Soc., Provi- dence, Rhode Island, 1964. 14. Lyndon, R. C. and Schupp, P. E. : Combinatorial Group Theory, Springer, Berlin, Heidel- berg, New York, 1977. 15. Magnus, W.: Noneuclidean Tesselations and their Groups, Academic Press, New York and London, 1974. 16. Maskit, B.: 'On Poincar6's Theorem for Fundamental Polygons', Advances in Math. 7 (1971), 219-230. 17. Mostow, G. D. : Strong Rigidity of Locally Symmetric Spaces, Princeton, New Jersey, 1973. 18. Sullivan, D. : 'On the Ergodic Theory at Infinity of an Arbitrary Discrete Group of Hyper- bolic Motions' in Proc. Stony Brook Conf. on Kleinian Groups and Riemann Surfaces, June, 1978. 19. Sullivan, D. : 'The Density at Infinity of a Discrete Group of Hyperbolic Motions', PubL Math. IHES 50 (1979), 171-202. 20. Thurston, W. P. : 'Geometry and Topology of 3-Manifolds', Notes from Princeton Univ., 1978. 21. Thurston, W. P. : 'On the Geometry and Dynamics of Diffeomorphisms of Surfaces, I (Preprint). 22. Thurston, W. P. : 'Hyperbolic Structures on 3-Manifolds, I : Deformation of Anannular Manifolds' (Preprint). Author's address: James W. Cannon, Mathematics Department, University of Wisconsin, 480 Lincoln Drive, Madison, W1 53706, U.S.A (Received January 28, 1981 ; revised version, December 19, 1983)