Boundaries and random walks on finitely generated infinite groups

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Ark. Mat., 41 (2003), 295 306

Boundaries and random walks on finitely generated infinite groups

A n d e r s K a r l s s o n

A b s t r a c t . We prove that almost every path of a random walk on a finitely generated non- amenable group converges in the compactification of the group introduced by W. J. Floyd. In fact, we consider the more general setting of ergodic cocycles of some semigroup of one-Lipschitz maps of a complete metric space with a boundary constructed following Gromov. We obtain in addition that when the Floyd boundary of a finitely generated group is non-trivial, then it is in fact maximal in the sense that it can be identified with the Poisson boundary of the group with reasonable measures. The proof relies on works of Kaimanovich together with visibility properties of Floyd boundaries. Furthermore, we discuss mean proximality of 0[' and a conjecture of McMullen. Lastly, related statements about the convergence of certain sequences of points, for example quasigeodesic rays or orbits of one-Lipschitz maps, are obtained.

1. I n t r o d u c t i o n

I n several s i t u a t i o n s c o n c e r n i n g a n infinite g r o u p it has b e e n useful to consider a n a u x i l i a r y space which in some sense is a b o u n d a r y . For a finitely g e n e r a t e d g r o u p F one could s t a r t w i t h t h e C a y l e y g r a p h K ( F , S) w i t h respect to some finite set of g e n e r a t o r s S. T h e

end-cornpactification

of t h e g r o u p is t h e g r a p h itself u n i o n t h e space of e n d s of this g r a p h a n d was i n t r o d u c e d b y H. F r e u d e n t h a l , see [S]. T h e r e are however c e r t a i n g r o u p s w i t h o n l y one end, b u t for which one would like to have a n o n - t r i v i a l b o u n d a r y . For e x a m p l e this is t h e case for f u n d a m e n t a l g r o u p s of c o m p a c t n e g a t i v e l y c u r v e d manifolds. A finer c o m p a c t i f i c a t i o n P FUcgF was first used b y W . a. F l o y d [F1] a n d it is o b t a i n e d by rescaling t h e l e n g t h one edges in a c e r t a i n way so t h a t t h e g r a p h gets finite d i a m e t e r , t h e n t a k i n g t h e c o m p l e t i o n of t h e g r a p h as a m e t r i c space. I n d e e d , such a b o u n d a r y 0 F of a f u n d a m e n t a l g r o u p F of a c o m p a c t surface of g e n u s at least two is t h e circle. S t a r t i n g w i t h IF1]

this c o m p a c t i f i c a t i o n has b e e n used i n K l e i n i a n g r o u p theory. I n this c o n t e x t , we wish to d r a w t h e r e a d e r ' s a t t e n t i o n to a c o n j e c t u r e s t a t e d i n a r e c e n t p a p e r by C. M c M u l l e n [Mc] c o n c e r n i n g t h e existence of a b o u n d a r y m a p from t h e F l o y d

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296 A n d e r s K a r l s s o n

b o u n d a r y of a fundamental group into the b o u n d a r y of hyperbolic three-space.

In this note we prove the convergence of quasigeodesics and p a t h s of certain r a n d o m walks in a geodesic space to points in a b o u n d a r y which is constructed following M. G r o m o v [G] extending Floyd. In particular, when F is a finitely generated non-amenable group with a measure p whose support generates the group, we obtain t h a t the Floyd b o u n d a r y OF is a / z - b o u n d a r y in the sense of H. Fursten- berg, see [Fu2] and [Ka2]. Using a different approach, by d e m o n s t r a t i n g certain visibility properties and then relying on work of V. Kaimanovich we obtain t h a t this p - b o u n d a r y is in fact either trivial or maximM; in the latter case it is the Pois- son boundary. In general, it m a y h a p p e n t h a t the b o u n d a r y is trivial: the Floyd b o u n d a r y of the product of two finitely generated infinite groups is one point ([F1]).

It was previously known t h a t the Poisson b o u n d a r y (for reasonable measures) of a group with infinitely m a n y ends can be identified with the space of ends with the hitting measure. See W. Woess [W], D. I. Cartwright and P. M. Soardi [CS], and Kaimanovich [Ka2]. Furthermore, Kaimanovich obtained an identification of the Poisson b o u n d a r y for hyperbolic groups, see [Kal] and INn2]. See also the work of A. Ancona [A].

Note also the somewhat related results of Floyd IF2] and C. W. Stark [St], which extend the result in the original p a p e r IF1] and concern the comparison of OF with the Furstenberg b o u n d a r y for rank one s y m m e t r i c spaces.

T h e results in this p a p e r were announced at the workshop on R a n d o m Walks and G e o m e t r y at the Erwin SchrSdinger Institute, Vienna, Summer 2001. I wish to t h a n k the organizers (Kaimanovich, Schmidt, Woess) and the Erwin SchrSdinger I n s t i t u t e tbr the financial support allowing me to participate in this stimulating conference.

2. P r e l i m i n a r i e s o n m e t r i c s p a c e s

T h e following material is standard; we borrow some notation from [BHK].

Let (Y,d) be a metric space. T h e length of a continuous curve a: [a, b ] ~ Y is defined to be

= sup Z

i = 1

where the s u p r e m u m is taken over all finite partitions a=to < t l < . . . < t k = b . W h e n this s u p r e m u m is finite, ct is said to be rectifiable. For such c~ we can define the arc

length s: [a, b]-+ [0, co) by

s(O = L(~I[~,,]), which is a function of bounded variation.

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B o u n d a r i e s a n d r a n d o m w a l k s o n f i n i t e l y g e n e r a t e d i n f i n i t e g r o u p s 2 9 7

A geodesic

is a curve fl for which

d(3(t),/~(t')) = L(Sl[t,,,]) = ] t - t ' l

for any t and t'. A metric space is called

geodesic

if any two points can be joined by a geodesic segment.

Given a continuous, (strictly) positive function f on Y, we define the

f-length

of a rectifiable curve a to be

Lf(a) = f ds = f(a(t)) ds(t).

If f - l , then

Lf =L.

Assume from now on t h a t (Y~d) is a geodesic space. A new distance df is defined by

df (x, y) = inf

Lf (c~),

where the infimum is taken over all rectifiable curves a with

a(a)=x

and

a(b)=y.

It is straightforward to verify t h a t

(Y, dr)

indeed is a metric space and t h a t the two metrics induce the same topology. Note t h a t for a geodesic/3, we have the simple bound

L f ( 9 ) _< L ( Z ) ~ 2 f ( x ) .

3. D e f i n i t i o n o f f - b o u n d a r i e s

This section defines certain boundaries of a complete metric space. T h e construction here is a somewhat more restrictive version of the one given by G r o m o v [G], Section 7.2.K "A confbrmal view on the b o u n d a r y " , which extends Floyd [F1], which in turn is "based on an idea of T h u r s t o n ' s and inspired by a construction of Sulli-

v a n ~ s ~ .

Let (Y, d) be a complete metric space which is geodesic and let f be a continuous, (strictly) positive function on Y. We will assume t h a t for some point y in Y, f is b o u n d e d by a m o n o t o n e real function F in the way

(F1) f @ ) <_ F ( d ( y , x ) ) for every co.

F u r t h e r m o r e we require t h a t this F is summable:

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r = l

and t h a t for every c > 0 there is a n u m b e r N such t h a t (F3)

F(cr) < NF(r)

for all r > 0.

Let the

f-boundary

of Y be the space

O f Y : = Y f - Y ,

where Yf denotes the metric space completion of (Y, dr).

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298 Anders Karlsson

Floyd's boundary

This is essentially the construction introduced in IF1]. Let F be a group generated by a finite set of elements S. Associated to S there is a left-invariant metric

(word metric) d

on F and a one-complex

(Cayley graph)

K ( F , S). T h e vertices of this g r a p h consist of the elements of F, and two vertices are connected by an (un- oriented) edge if they differ by an element of S on the left. W h e n the edges are assigned to have length one, the distance d on F is simply the geodesic distance in the graph.

Let F be a morfic, s u m m a b l e function F : N - - + R , such t h a t given k E N there exists L, M, N > 0 so t h a t

MF(r') <_F(kr)<_NF(r)

and L - i f ( r )

<_f(s) <Lf(r)

for all natural numbers r and

r - k < s < r + k .

(It is c o m m o n to consider F ( r - 1 ) : = r - 2 . ) We insist for convenience t h a t F is monotonically decreasing and let f ( z ) =

F(d(z,e)).

Since F is s n m m a b l e the graph now has finite diameter. T h e

group completion in the sense of Floyd

F = F U O F is (just as above) the completion of the Cayley g r a p h with the new distance d I as a metric space (i.e. the equivalence classes of Cauchy sequences). The group F acts on F by homeomorphisms. If 0F consists of only zero, one, or two points, we say t h a t the b o u n d a r y is trivial.

Examples

It is easy to see t h a t there is a surjection from OF to the space of ends. There- fore, groups with infinitely m a n y ends have a non-trivial Floyd boundary. W h e n F is a word hyperbolic group then, under some conditions, the f - b o u n d a r y of F co- incides with the s t a n d a r d hyperbolic boundary, see [G], or [CDP] for an exposition with more details. T h e conjecture stated in [Mc] predicts t h a t the Floyd boundary of a finitely generated fundamental group of a hyperbolic three-manifold is at least as large as the limit set. For geometrically finite Kleinian groups (in every dimension) this was already proven in IF1] and IT1].

It was suggested by G r o m o v t h a t a Floyd type b o u n d a r y of a H a d a m a r d space (CAT(0)-space) consists of the set of Tits components of the usual ray boundary.

Let Y be a b o u n d e d convex domain equipped with Hilbert's metric and consider its n a t u r a l extrinsic b o u n d a r y

O ~ Y : = Y \ Y c R N.

Some calculations of G. Noskov and the author indicate t h a t a Floyd type b o u n d a r y is 0~Y modulo collapsing faces

t o points and identifying adjacent faces.

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Boundaries and random walks on finitely generated infinite groups 299

4. Ergodic cocycles and p-boundaries Let (Y,d), y, f, and OY:=OfY be as in Section 3.

Let (X, u) be a measure space with ~ ( X ) = I and let L: X - + X be an ergodic measure-preserving transformation. Let w: X--+S be a measurable m a p into a semigroup S of selfmaps of Y which does not increase d-distances. Assmne t h a t the integrability condition

j/x d(y, w(x)y) d . ( x ) < ^{O 0}

holds. Define the associated ergodic cocyele (or ~nndom product)

= w ( L

and

A := ,,-~lim

-nl ; d(e, u(n, x)) dp(x).

A basic observation is t h a t a(n,x):=d(y, u(n,x)y) is a subadditive cocycle. T h e following purely subadditive ergodic s t a t e m e n t was proved in [KM].

L e m m a . For each c>O, let Ee be the set of x for which there exist an integer K = K ( x ) and infinitely many n such that

(t)

a(n, x ) - a ( n - k , Lkx) > ( A - s ) k

for all k, K < k < n . Then u ( ~ > 0 E ~ ) = l .

Using this l e m m a we can prove the tbllowing result.

T h e o r e m . Assume that A > 0 . Then for almost every x the trajectory u(n, x)y converges to a point ~ = ~ ( x ) C O Y .

Proof. Fix an xEE~ for some s < A . For each m denote by Ym the point u(rn, x)y. Consider an ni and k, K<k<n~, such t h a t (t) in the l e m m a holds, and let/3 be a geodesic joining Yn~ and Yk. Let J be the smallest integer larger t h a n d(y,,~, yk) and J0 be the smallest integer larger t h a n 1 / r a i n { A - e , 89 In the ease Y = l we have t h a t

d f ( y ~ , yk) = inf" Lf(a) < m a x f(/3(t) )d(y,~.~, Yk)

oo

Z

r k j o

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300 Anders Karlsson

where N corresponds to c = A - c - 1 / K in (F3). In the case J > l , note that ~_< 1

d(y~, yk)/J< 1 and let

= ~ d(w~, w).

t5

Using the monotonicity of F, (F1), the triangle inequality, the inequality (t), and (F3) we have

d f ( y ~ , Yk) ~_ Lf(fl)

J

~ E nlax , f ( ~ ( t ) ) d ( / ~ ( ~ j 1), fi@j)) j = l t3-1~t<--td

J

-< Z r(d(y, Z(tj))- ~)

j = l J

< E F ( a ( n i , x ) _ ( d ( y ~ , y ~ ) jd(y,~,yk)) 1)

j-1 - J

J

< ~ ( A - c ) k + ~ -

-j~-i

⁽ ^{z 1)}

J

< X~F(j+~-jo)

j--1

( x 3

_<N E r(r).

r=k--jo

Since N and j0 depend only on A - e , and because of (F2), it follows that u(ni, x)y is a Cauchy sequence and moreover that the whole sequence u(m, x)y hence converges to a point in 0Y. []

We also have the following consequence.

C o r o l l a r y . Let F be a finitely generated group and tt be a probability measure with finite first moment such that the support of It generates F as a semigroup.

Assume that the Poisson boundary of (F, It) is non-trivial (which is the case if F is non-amenable). Then almost every sample path of the random walk (F, p) converges to a (random) point of OF (which is defined as in the example in Section 3). The space OF with the resulting limit measure is a It-boundary.

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Proof.

We will a p p l y the t h e o r e m to (Y, d) being the group F equipped with a left-invariant word-distance. Furthermore, we let S = F , which acts on Y = F by translations preserving the word-metric. Let (X, v) be the infinite p r o d u c t of (F, p) and L be the shift. It is a known fact t h a t L is an ergodic measure preserving transformation, and the finiteness of the first m o m e n t simply translates into the integrability assumed above.

It is known t h a t when the Poisson b o u n d a r y is non-trivial, the word length of the t r a j e c t o r y grows linearly (so A > 0 ) for almost every trajectory, see [Gu] or T h e o r e m 5.5 in [Ka2]. Hence by the theorem, almost every sample p a t h converges to a point in OF.

T h e resulting measure space, 0F with the hitting measure, is a p-boundary, see [Ful] or, e.g., [Ka2], p. 660. []

Note t h a t it does not seem to be clear whether the p - b o u n d a r y obtained is non-trivial or not. In the next sections however, by combining some observations about the Floyd b o u n d a r y with works of Kaimanovieh, we obtain t h a t whenever the Floyd b o u n d a r y is non-trivial, then it is in fact maximal.

5. Visibility of Floyd's boundary

Let F be a finitely generated infinite group with a b o u n d a r y OF of Floyd type, see the example in Section 3. We s t a r t with a lemma.

L e m m a .

Let z and w be two points in F and let [z, w] be a geodesic segment connecting z and w. Then

~o

d'(z, < 4rY(,-)+2 r(j),

j = r

wher [z,

Proof.

Let a denote the distance to z from a point m on [z, w] closest to e.

Let

xj,

j = 0 , . . . ,a, be the points (vertices) of the geodesic segment [m,

z]C[w,z].

Because of the minimality of r and the triangle inequality we have the estimates

d(e, xj) k r

and

d(e, x j ) > j - r .

For the usual reasons, we hence get

d'(m, z) < f i F(min{d(e, xj), d(e,

x j + l ) } )

j-o

2r--1 f i oo

<- E F ( r ) +

F(j r ) < _ 2 r F ( r ) + E F ( j ).

j--0 j--2r j--r

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(If a < 2 r - 1 , then we would not decompose the sum, and so the second t e r m would not be present.) By the same consideration with w instead of z, the l e m m a is proved. []

Note t h a t with only minor modifications, this proof works for any f - b o u n d a r y of a geodesic space Y.

Two b o u n d a r y points 7 and ~ are said to be connected by a geodesic line if there is a d-geodesic a such t h a t ~(n)-+-y and a ( - n ) - + ~ , as n-+cxD.

P r o p o s i t i o n . Every two points in OF can be connected by a geodesic line.

Proof. It is known and easy to show (see [F1]) t h a t every point 3,~c9F can be represented as an endpoint of a geodesic ray from e. Consider the geodesic segments [~/(n), ((n)] for any two distinct b o u n d a r y points. It follows from the l e m m a t h a t the distance r , from these curve segments to e must be bounded (due to the s u m m a b i l i t y of F and since 7 ( o 0 ) # { ( o o ) ) . T h a n k s to the local finiteness of F (the Cayley graph) we m a y now extract a desired geodesic line using C a n t o r ' s diagonal argument. []

6. P o i s s o n b o u n d a r i e s

T h e results in the previous section lead to (in notation and definitions as in [Ka2]) the following theorem.

T h e o r e m . If OF contains at least three points, then the compactification F of Floyd type satisfies Kaimanovich's conditions (CP), (CS), and (CC).

Proof. It is known t h a t the (isometric) action of F by left translation on itself extends to an action on F by homeomorphisms, see for example [K2]. Since [' is a compact metric space it is separable. Any two sequences of bounded distance fl'om each other clearly cannot converge to different b o u n d a r y points; this is (CP).

As shown in tile previous section, any two b o u n d a r y points c~n be joined by d-geodesics. Therefore the sets (strips) S('y, ~) which equal the union of ail joining geodesic lines are n o n - e m p t y and constitute an equivariant family of Borel maps.

Furthermore, for any three distinct b o u n d a r y points (i, i = 1 , 2~ 3, we m a y find small neighborhoods U~ in F of each of the three points so t h a t

s(u~, u 2 ) n u 3 = r

To see this, take for O~i small disjoint c-neighborhoods in the metric d'. Now assume t h a t we can find geodesic lines O~k in S(U1, Us), p a r a m e t r i z e d so t h a t 7~(0) is a point closest to e and indices n k - + o o (because every geodesic line in S intersects

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a bali around e) such t h a t 7k(nk)--+ ~, where ~ is some point in UaNOF. But then, since the dr-length of every d-geodesic ray is unitbrmly bounded, it follows t h a t d' (Tk (nk), "Yk (o c))--+ 0, and this cannot h a p p e n because the neighborhoods were disjoint. This proves (CS).

T h e condition (CG): d is a left-invariant metric on F and the corresponding gauge is t e m p e r a t e (just meaning in this context t h a t F is finitely generated). Fi- nally, as already discussed, tbr any two b o u n d a r y points every joining geodesic line intersects the same d-finite radius ball around e (a consequence of the lamina in Section 5). []

We can now invoke the nice argmnents of Kaimanovich in [Ka2], Sections 2, 3, 4, and 6, which in particular involve entropy considerations of certain conditional r a n d o m walks (the strip approximation criterion) as well as a version of the martin- gale convergence t h e o r e m ([Fu2]). Corresponding to [Ka2, T h e o r e m 6.6] we hence have the following result.

C o r o l l a r y . Let F be a finitely generated g~vup and assume that a Floyd type boundary OF contains more than three points. Let p be a probability measure on F with finite entropy and finite first logarithmic moment, and whose support generates a subgrmtp which is non-elementary with respect to F, that is, it does not fiz a finite subset of OF. Then the cornpactification F is #-maximal.

T h e Floyd b o u n d a r y of a firlitely generated amenable group consists of zero, one, or two points since otherwise the group contains n o n - c o m m u t a t i v e free subgroups ([K2]). There are however solvable groups for which ti~e Poisson b o u n d a r y is non-trivial fbr any (non-degenerate) probability measure with finite entropy (due to Kairnanovich, u and Erschler).

7. M e a n p r o x i m a l i t y

Let us record some fln'ther consequences of tile previous two sections. For the definitions and a basic account of Furstenberg's b o u n d a r y theory, we refer to tile original source [Fu2] as well as [M, C h a p t e r VII.

T h e o r e m . Let F be a finitely generated grwup and assume that Or is a non- trivial Floyd boundary. Then Or is a mean proximal X-space.

Proof. In [K21 it is shown t h a t 0F is a b o u n d a r y of" F in the sense of [Fu2].

Therefore, in particular, F does not leave a finite subset of OF invariant. In view of this, tile t h e o r e m in Section 6, and the elegant arguments of Kaimanovich in [Ka2, Section 2.4], we have t h a t OF is mean proximal. []

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If the Floyd b o u n d a r y is non-trivial for a finitely generated fundamental group of a hyperbolic three-manifold, then by knowing t h a t it can be identified with the Poisson b o u n d a r y as we now do (Section 6), we hence get a unique (up to null sets) F-equivariant measurable m a p onto the limit set. (This is part of the theory of Poisson boundaries, see [Ka2]; for the uniqueness we refer in addition to [M, C h a p t e r VI, Corollary 2.1010 W h e n does this m a p agree a.e. with a continuous m a p ? This question, together with the issue of the non-triviality of OF, is in a sense now the content of the conjecture stated in [Mc].

Added in pro@ T h e non-triviality of OF is in fact not an issue: Let F be a finitely generated fundamental group of a hyperbolic three-manifold. It is known from Scott's core t h e o r e m and T h u r s t o n ' s uniformization t h e o r e m t h a t the group F can be m a d e to act as a geometrically finite Kleinian group, see [E], pp. 266 267. Hence we know from IF1] t h a t the b o u n d a r y OF is infinite (provided t h a t F is non-elementary of course) and we have.

T h e o r e m . Let N be a hyperbolic three-manifold with finitely generated funda- mental group and denote by A its limit set on the boundary of the hyperbolic three- space. Then there exists a unique (up to null sets) measurable, 7h(N)-eqnivariant map

F : c~7c 1 ( i V ) ---} A .

It is perhaps interesting to compare this result with [T2], which also considers measurable b o u n d a r y maps. In view of contractivity properties, quite generally, a m a p such as F either agrees with a continuous m a p or is very discontinuous (the F - i m a g e of every neighborhood of a point is dense in A). For references to other papers discussing the conjecture, see [Mc].

8. C o n v e r g e n c e o f c e r t a i n s e q u e n c e s

Let (Y, d), y, f , and cqY:=OfY be as in the previous section.

P r o p o s i t i o n . Let y,~ be a sequence of points in Y for which there exists two positive constants A and C such that d(y~, y,~+l) < C and d(y,,~, y) > A n for all large n. Then y~ converges to a point in OY.

Prvof. Let/3 be a geodesic joining y~ and Y~+I. For every large n we have

i fLs(-) <L t(9)

_< f(9(t)) d(y..,

< r(A -C)C <

t

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Hence the sequence of points in question is a d~-Cauchy sequence, because F is summable, and N and C are independent of n. As y.~--+oc in Y, the sequence therefore converges to a point in the b o u n d a r y of Y. []

As a corollary of the proposition we have that any quasigeodesic ray converges.

If F ( r - 1 ) = r -(2+~), for some positive E, then a similar argument also shows that any regular sequence in the sense of [Ka2] converges.

Using the argument in the proof of Proposition 5.1 in [K1], the lemma in Section 5 and the inequality

<_ d(v, b',

one can prove the following proposition.

P r o p o s i t i o n . Let r be a one-Lipschitz map of a complete, geodesic metric space Y and let Of Y be an f-boundary. If d(6~(y),y)--+oo, then ~ ( y ) converges to a point in Of Y , as n--+ec.

Let us emphasize that here, as well as in some other statements in this paper, no local compactness is assumed.

R e f e r e n c e s

[A] ANCONA, A., Th6orie du poteutiel sin" les graphes et les vari6t6s, in Ecole d'dtd de Probabilit& de Saint-Flour XVIII, 1988 (Hennequin, P. L., ed.), Lecture Notes in Math. 1427, pp. 1-112, Springer-Verlag, Berlin Heidelberg, 1990.

[BHK] BONK, M., HEINONEN, J. and KOSKELA, P., Uniformizing Gromov hyperbolic space, Astdrisque 270 (2001), 1-99.

[CS] CARTWRIGH% D. I. and SOARDI, P. M., Convergence to ends for random walks on the automorphism group of a tree, Prvc. Amer. Math. Soc. 107 (1989), 817-823.

[CDP] COORNAER~, M., DELZANT, T. and PAPADOPOULOS, A., Gdomdtrie et thdorie des groupes. Les groupes hyperboliques de Gromov. Lecture Notes in Math.

1441, Springer-Verlag, Berlin Heidelberg, 1990.

[El EPSTEIN, D. B. A., CANNON, J. W., HOLT, D. F., LEVY, S. V. F., PATERSON, M.

S. and THURSTON, W. P., Word Processing in Groups, Jones and Bartlett, Boston, Mass., 1992.

IF1] FLOYD, W. J., Group completions and limit sets of Kleinian groups, Invent. Math.

57 (1980), 205-218.

IF2] FLOYD, W. J., Group completions and Furstenberg boundaries: rank one, Duke Math. J. 51 (1984), 1017 1020.

[Ful] FURSTENBEI~O, H., Random walks and discrete subgroups of Lie groups, in Ad- vances in Probability and Related Topics, vol. 1 (Ney, P., ed.), pp. 1 63, Marcel Dekker, New York, 1971.

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306 Anders Karlsson: Boundaries and random walks on finitely generated infinite groups

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Received December 1Z 2001 Anders Karlsson

Department of Mathematics Royal Institute of Technology SE-100 44 Stockholm

Sweden

emaih karlsson~aya.yale.edu