Hyperbolic spaces from self-similar group actions

We show that the limit space of a contracting selfsimilar group action is the boundary of a naturally defined Gromov hyperbolic space.

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Дата:	2003
Автор:	Nekrashevych, V.
Формат:	Стаття
Мова:	English
Опубліковано:	Інститут прикладної математики і механіки НАН України 2003
Назва видання:	Algebra and Discrete Mathematics
Онлайн доступ:	http://dspace.nbuv.gov.ua/handle/123456789/154679
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Цитувати:	Hyperbolic spaces from self-similar group actions / V. Nekrashevych // Algebra and Discrete Mathematics. — 2003. — Vol. 2, № 1. — С. 77–86. — Бібліогр.: 14 назв. — англ.

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spelling	irk-123456789-1546792019-06-16T01:31:40Z Hyperbolic spaces from self-similar group actions Nekrashevych, V. We show that the limit space of a contracting selfsimilar group action is the boundary of a naturally defined Gromov hyperbolic space. 2003 Article Hyperbolic spaces from self-similar group actions / V. Nekrashevych // Algebra and Discrete Mathematics. — 2003. — Vol. 2, № 1. — С. 77–86. — Бібліогр.: 14 назв. — англ. 1726-3255 2001 Mathematics Subject Classification: 37B05, 37D99. http://dspace.nbuv.gov.ua/handle/123456789/154679 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	We show that the limit space of a contracting selfsimilar group action is the boundary of a naturally defined Gromov hyperbolic space.
format	Article
author	Nekrashevych, V.
spellingShingle	Nekrashevych, V. Hyperbolic spaces from self-similar group actions Algebra and Discrete Mathematics
author_facet	Nekrashevych, V.
author_sort	Nekrashevych, V.
title	Hyperbolic spaces from self-similar group actions
title_short	Hyperbolic spaces from self-similar group actions
title_full	Hyperbolic spaces from self-similar group actions
title_fullStr	Hyperbolic spaces from self-similar group actions
title_full_unstemmed	Hyperbolic spaces from self-similar group actions
title_sort	hyperbolic spaces from self-similar group actions
publisher	Інститут прикладної математики і механіки НАН України
publishDate	2003
url	http://dspace.nbuv.gov.ua/handle/123456789/154679
citation_txt	Hyperbolic spaces from self-similar group actions / V. Nekrashevych // Algebra and Discrete Mathematics. — 2003. — Vol. 2, № 1. — С. 77–86. — Бібліогр.: 14 назв. — англ.
series	Algebra and Discrete Mathematics
work_keys_str_mv	AT nekrashevychv hyperbolicspacesfromselfsimilargroupactions
first_indexed	2025-07-14T06:42:24Z
last_indexed	2025-07-14T06:42:24Z
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fulltext	Jo ur na l A lg eb ra D is cr et e M at h.Algebra and Discrete Mathematics RESEARCH ARTICLE Number 2. (2003). pp. 77–86 c© Journal “Algebra and Discrete Mathematics” Hyperbolic spaces from self-similar group actions Volodymyr Nekrashevych Communicated by V. I. Sushchansky Abstract. We show that the limit space of a contracting self- similar group action is the boundary of a naturally defined Gromov hyperbolic space. 1. Introduction Self-similar group actions (or self-similar groups) have proved to be in- teresting mathematical objects from the point of view of group theory and many other fields of mathematics (operator algebras, holomorphic dynamics, automata theory, etc). See the works [BGN02, GNS00, Gri00, Sid98, BG00, Nek02a], where different aspects of self-similar groups are studied. An important class of self-similar group actions are contracting ac- tions. Contracting groups have many nice properties. For example, the word problem is solvable in a contracting group in a polynomial time [Nek]. The author has shown (see [Nek02b]) that a naturally de- fined topological space, called the limit space, is associated with every contracting self-similar action. This topological space is metrizable and finite-dimensional. It was discovered later (see [Nek02a]) that with many topological dy- namical systems (like iterations of a rational function) a self-similar group The research was supported by the Swiss National Science Foundation and Alexan- der von Humboldt Foundation 2001 Mathematics Subject Classification: 37B05, 37D99. Key words and phrases: self-similar group actions, limit space, Gromov hy- perbolic spaces, boundary of a hyperbolic space. Jo ur na l A lg eb ra D is cr et e M at h.78 Hyperbolic spaces from self-similar group actions is associated. This self-similar group (called the iterated monodromy group) is often contracting and in fact contains all the essential dynam- ics of the original dynamical system. In particular, if the dynamical system is expanding, then its iterated monodromy group is contracting and the limit space of the iterated monodromy group is homeomorphic to the Julia set of the dynamical system. We show in this paper how to associate with every self-similar action a graph, which will be Gromov-hyperbolic if the group action is con- tracting. The boundary of this graph is then homeomorphic to the limit space of the group action. In some sense this result (together with the notion of iterated mon- odromy group) can be interpreted as a new entry to the “Sullivan dic- tionary” [Sul85], which shows an analogy between the holomorphic dy- namics and groups acting on hyperbolic spaces. The author is grateful to L. Bartholdi, R. Grigorchuk, V. Kaimanovich for helpful discussions on the subject of the paper. 2. Self-similar actions Let X be a finite set (alphabet). By X∗ we denote the set of all finite words over the alphabet X, including the empty one ∅. By X−ω we denote the space of all infinite sequences to the left of the form . . . x2x1, with the topology of the infinite countable direct power of the discrete set X. Definition 2.1. A faithful action of a group G on the set X∗ is said to be self-similar if for every g ∈ G and every x ∈ X there exist h ∈ G such that g(xw) = g(x)h(w) (1) for every w ∈ X∗. Iterating equation (1), we get that for every v ∈ X∗ there exists an element h such that g(vw) = g(v)h(w). The element h is uniquely defined and is called restriction of the element g in the word v and is denoted g\|v. We easily get the following properties of restrictions: g\|v1v2 = g\|v1 \|v2 (g1g2)\|v = ( g1\|g2(v) ) (g2\|v) . (2) Definition 2.2. Let Pn : (Xn)∗ −→ X∗ be the injective map, which carries the word (x1, x2, . . . , xn)(xn+1, xn+2, . . . , x2n) . . . . . . (x(k−1)n+1, x(k−1)n+2, . . . , xkn) ∈ (Xn)∗ Jo ur na l A lg eb ra D is cr et e M at h.V. Nekrashevych 79 to the word x1x2x3 . . . xkn ∈ X∗. The nth tensor power of a self-similar action is the action on the set (Xn)∗ obtained by conjugating the action on X∗ by the map Pn. The nth power of a self-similar action obviously is also self-similar. The adding machine. One of the most classical examples of a self- similar action is the “adding machine” defined in the following way. We put X = {0 , 1} and define a transformation a recurrently by the rules a(0w) = 1w a(1w) = 0a(w) for all w ∈ X∗. It is easy to prove that if an(x1x2 . . . xm) = y1y2 . . . ym then y1 + y2 · 2 + y3 · 2 2 + y4 · 2 3 + . . . ym · 2m−1 = ( x1 + x2 · 2 + x3 · 2 2 + x4 · 2 3 + . . . xm · 2m−1 ) + n (mod 2m). 3. Contracting actions and their limit spaces Definition 3.1. A self-similar action of a group G is called contracting if there exists a finite set N ⊂ G such that for every g ∈ G there exists k ∈ N such that g\|v ∈ N for all the words v ∈ X∗ of length ≥ k. The minimal set N with this property is called the nucleus of the self-similar action. As an example of a contracting action, one can take the adding ma- chine action. Other examples of contracting actions include the Grig- orchuk group and the iterated monodromy groups of post-critically finite rational functions. Let us fix some contracting self-similar action of a group G on the set X∗. Definition 3.2. Two sequences . . . x3x2x1, . . . y3y2y1 ∈ X−ω are said to be asymptotically equivalent with respect to the action of the group G, if there exist a finite set K ⊂ G and a sequence gk ∈ K, k ∈ N such that gk(xkxk−1 . . . x2x1) = ykyk−1 . . . y2y1 for every k ∈ N. Definition 3.3. The limit space of the self-similar action (denoted JG) is the quotient of the topological space X−ω by the asymptotic equivalence relation. Jo ur na l A lg eb ra D is cr et e M at h.80 Hyperbolic spaces from self-similar group actions Example. In the case of the adding machine action of Z one can prove (see [Nek02b]) that two sequences are asymptotically equivalent if and only if ∞ ∑ n=1 xn · 2−n = ∞ ∑ n=1 yn · 2−n (mod 1). Consequently, the limit space JZ is the circle R/Z. We have the following properties of the limit spaces, which are proved in [Nek02b]. Proposition 3.1. The limit space JG is metrizable and has topological dimension ≤ \|N \| − 1, where N is the nucleus of the action. 4. The limit space as a hyperbolic boundary Definition 4.1. Let G be a finitely generated group with a self-similar action on X∗. For a given finite generating system S of the group G we define the self-similarity graph Σ(G, S) as the graph with the set of vertices X∗ and two vertices v1, v2 ∈ X∗ belonging to a common edge if and only if either vi = xvj for some x ∈ X (the vertical edges) or s(vi) = vj for some s ∈ S (the horizontal edges), where {i, j} = {1, 2}. As an example, see a part of the self-similarity graph of the adding machine on Figure 1. Figure 1: The self-similarity graph of the adding machine If all restriction of the elements of the generating set S also belong to S, then the self-similarity graph Σ(G, S) is an augmented tree in sense of V. Kaimanovich (see [Kai03]). Jo ur na l A lg eb ra D is cr et e M at h.V. Nekrashevych 81 We have a natural metric on the set of the vertices of any graph. The distance between two vertices in this metric is equal to the number of edges in the shortest path connecting them. The definition of the self-similarity graph depends on the choice of the generating set S. We will use the classical notion of quasi-isometry in order to make it more canonical. Definition 4.2. Two metric spaces X and Y are said to be quasi- isometric if there exists a map (which is called then a quasi-isometry) f : X −→ Y and positive constants L, C such that (i) L−1dX(x1, x2) − C < dY(f(x1), f(x2)) < LdX(x1, x2) + C, for all x1, x2 ∈ X and (ii) for every y ∈ Y there exists x ∈ X such that dY(y, f(x)) < C. Lemma 4.1. Let G be a group with a self-similar action, and let φ be the associated virtual endomorphism. 1. The self-similarity graphs Σ(G, S1) and Σ(G, S2), where S1, S2 are two different finite generating sets of the group G, are quasi- isometric. 2. The self-similarity graph of the nth tensor power of the self-similar action is quasi-isometric to the self-similarity graph of the original action. Proof. 1) The identical map on the set of vertices Σ(G, S1) −→ Σ(G, S2) is a quasi-isometry. The constant L is any number such that the length of every element of one of the generating sets has length less than L with respect to the other generating set. The constant C can be any positive number. 2) The set of vertices of the nth tensor power is equal to {∅} ∪Xn ∪ X2n ∪ X3n ∪ . . .. Let F : Σn(G, S) −→ Σ(G, S) be the natural inclusion of the vertex sets. It is easy to see that d(F (u), F (v)) ≤ n · d(u, v) and d(F (u), F (v)) ≥ d(u, v) for all u, v ∈ Σ(G, S). For every vertex v = x1x2 . . . xm ∈ X∗ of the graph Σ(G, S) there exists a vertex xrxr+1 . . . xm belonging to the vertex set of the graph Σn(G, S), which is at the distance less than n from v (one must take r to be the minimal number, such that m− r + 1 is divisible by n). So the map F satisfies both conditions of Definition 4.2. Jo ur na l A lg eb ra D is cr et e M at h.82 Hyperbolic spaces from self-similar group actions Let us recall the definition of Gromov-hyperbolic metric spaces [Gro87]. Let X be a metric space with the metric d(·, ·). The Gromov product of two points x, y ∈ X with respect to the basepoint x0 ∈ X is the number 〈x · y〉 = 〈x · y〉x0 = 1 2 (d (x, x0) + d (y, x0) − d (x, y)) . Definition 4.3. A metric space X is said to be Gromov-hyperbolic (see [Gro87]) if there exists δ > 0 such that the inequality 〈x · y〉 ≥ min (〈x · z〉 , 〈y · z〉) − δ (3) holds for all x, y, z ∈ X. If a proper geodesic metric space (for instance a graph) is quasi- isometric to a hyperbolic space, then it is also hyperbolic. For proofs of the mentioned facts and for other properties of the hyperbolic spaces and groups look one of the books [Gro87, CDP90, GH90]. Theorem 4.2. If the action of a finitely-generated group G is contracting then the self-similarity graph Σ(G, S) is a Gromov-hyperbolic space. Proof. It is sufficient to prove that some quasi-isometric graph is hyper- bolic. Therefore, we can change by statement (1) of Lemma 4.1 the set of generators S so that it will contain all the restrictions of its elements and that there exists N ∈ N such that for every element g ∈ G of length ≤ 4 and any word x1x2 . . . xN ∈ X∗, the restriction g\|x1x2...xN belongs to S. Then the length of any restriction of an element g ∈ G is not greater then the length of g. It is sufficient, by statement (3) of Lemma 4.1, to prove that the self-similarity graph ΣN (G, S) of the Nth tensor power of the action is Gromov-hyperbolic. Let us prove the following lemma. Lemma 4.3. Any two vertices w1, w2 of the graph ΣN (G, S) can be writ- ten in the form w1 = a1a2 . . . anw, w2 = b1b2 . . . bmg(w), where ai, bi ∈ XN , w ∈ ( XN )∗ , g ∈ G, l(g) ≤ 4 and d(w1, w2) = n + m + l(g). Then the Gromov product 〈w1 · w2〉 with respect to the basepoint ∅ is equal to \|w\| − l(g)/2. Here l(g) denotes the length of the element g with respect to some fixed finite generating set of the group. Proof. Let v1 = w1, v2, . . . vk = w2 be the consecutive vertices of the shortest path connecting the vertices w1 and w2. Then every vi+1 is obtained from vi by application of one of the following procedures: Jo ur na l A lg eb ra D is cr et e M at h.V. Nekrashevych 83 1. deletion of the first letter a ∈ XN in vi (descending edges); 2. adding a letter a ∈ XN to the beginning of vi (ascending edges); 3. application of an element of S to vi (horizontal edges); If the path has three consecutive vertices vi, vi+1, vi+2 such that vi+1 = avi, a ∈ XN and vi+2 = s(vi+1) for s ∈ S then vi+2 = bs′(vi), where b = s(a) ∈ XN and s′ = s\|a ∈ S. We replace the segment {vi, vi+1, vi+2} of the path by the segment {vi, s ′(vi), bs ′(vi) = vi+2}. If the path has three consecutive vertices vi, vi+1, vi+2 such that vi+1 = s(vi) for s ∈ S and vi+1 = avi+2 then vi = s−1(avi+2) = bs′(vi+2), where b = s−1(a) ∈ XN and s′ = s−1\|a ∈ S. Then we replace the segment {vi, vi+1, vi+2} of the path by the segment {vi = bs′(vi+2), s ′(vi+2), vi+2}. Let us perform these replacements as many times as possible. Then we will not change the length of the path, so each time we will get a geodesic path connecting the vertices w1, w2. Note that a geodesic path can not have a descending edge next after an ascending one. Therefore, eventually after a finite number of replacements we will get a geodesic path in which we have at first only descending, then horizontal and then only ascending edges. Then w1 = a1a2 . . . anw, w2 = b1b2 . . . bmg(w), with ai, bi ∈ XN , w ∈ ( XN )∗ , g ∈ G, and d(w1, w2) = n + m + l(g). Suppose that l(g) > 4. Let w = aw′, a ∈ XN and denote b = g(a) and h = g\|a. Then by the choice of the number N we have l (h) ≤ l(g) − 3. Since w1 = a1a2 . . . anaw′ and w2 = b1b2 . . . bmbh(w′), we have d(w1, w2) ≤ n + 1 + m + 1 + l(h) ≤ n + m + l(g) − 1, which contradicts to the fact that the original path was the shortest one. We have 〈w1 · w2〉 = 1 2 (n + \|w\| + m + \|w\| − (n + m + l(g))) = \|w\| − l(g) 2 . Let us take three points w1, w2, w3. We can write them by Lemma 4.3 as w1 = a1a2 . . . anw, w2 = b1b2 . . . bmg1(w) and w2 = b1b2 . . . bpu, w3 = c1c2 . . . cqg2(u), where ai, bi, ci ∈ XN , g1, g2 ∈ G, l(g1), l(g2) ≤ 4 and 〈w1 · w2〉 = \|w\| − l(g1)/2, 〈w2 · w3〉 = \|u\| − l(g2)/2. Jo ur na l A lg eb ra D is cr et e M at h.84 Hyperbolic spaces from self-similar group actions We can assume that p ≤ m. Then \|u\| ≥ \|w\| = \|g1(w)\|, so we can write u = vg1(w) for some v ∈ ( XN )∗ . Then w3 = c1c2 . . . cqg2(v)hg1(w), where h = g2\|v. We have l(h) ≤ l(g2) ≤ 4 and d(w1, w3) ≤ n + l(h) + l(g1) + q + \|v\|, hence 〈w1 · w3〉 = 1 2 (n + \|w\| + q + \|v\| + \|w\| − d(w1, w3)) ≥ ≥ \|w\| − (l(h) + l(g1))/2 ≥ \|w\| − 4. Finally, min(〈w1 · w2〉, 〈w2 · w3〉) ≤ 〈w1 · w2〉 ≤ \|w\|, so 〈w1 · w3〉 ≥ min(〈w1 · w2〉, 〈w2 · w3〉) − 4, and the graph ΣN (G, S) is 4-hyperbolic. Let X be a hyperbolic space. We say that a sequence {xn} of points of X converges to the infinity if the Gromov product 〈xn · xm〉 trends to infinity when m, n → ∞. This definition does not depend on the choice of the basepoint. We say that two sequences {xn} and {yn}, convergent to the infinity, are equivalent if limn,m→∞〈xn · ym〉 = ∞. The set of the equivalence classes of the sequences convergent to the infinity in the space X is called the hyperbolic boundary of the space X and is denoted ∂X. If a sequence {xn} converges to the infinity, then its limit is the equivalence class a ∈ ∂X, to which belongs {xn} and we say that {xn} converges to a. If a, b ∈ ∂X are two points of the boundary, then their Gromov product is defined as 〈a · b〉 = sup {xn}∈a,{ym}∈b lim inf m,n→∞ 〈xn · ym〉. For every r > 0 define Vr = {(a, b) ∈ ∂X × ∂X : 〈a · b〉 ≥ r}. Then the set {Vr : r ≥ 0} is a fundamental neighborhood basis of a uniform structure on ∂X (see [Bou71] for the definition of a uniform structure and [GH90] for proofs). We introduce on the boundary ∂X the topology defined by this uniform structure. Theorem 4.4. The limit space JG of a contracting action of a finitely generated group G is homeomorphic to the hyperbolic boundary ∂Σ(G, S) of the self-similarity graph Σ(G, S). Moreover, there exists a homeomor- phism F : JG −→ ∂Σ(G, S), such that D = F ◦ π, were π : X−ω −→ JG is the canonical projection and D : X−ω −→ ∂Σ(G, S) carries every sequence . . . x2x1 ∈ X−ω to its limit lim n→∞ xnxn−1 . . . x1 ∈ ∂Σ(G, S). Jo ur na l A lg eb ra D is cr et e M at h.V. Nekrashevych 85 Sketch of the proof. We will need the following well known result (see [CDP90] Theorem 2.2). Lemma 4.5. Let X1, X2 be proper geodesic hyperbolic spaces and let f1 : X1 −→ X2 be a quasi-isometry. Then a sequence {xn} of points of X1 converges to infinity if and only if the sequence {f1(xn)} does. The map ∂f1 : {xn} 7→ {f1(xn)} defines a homeomorphism ∂f1 : ∂X1 −→ ∂X2 of the boundaries. We pass, using Lemma 4.5, to an Nth tensor power of the self-similar action in the same way as in the proof of Theorem 4.2, so that for every g ∈ G such that l(g) ≤ 4 and for every a ∈ XN the restriction g\|a belongs to the generating set. Suppose that the sequence {wn} converges to the infinity. Choose its convergent subsequence in X−ω∪X∗. Suppose its limit is . . . x2x1 ∈ X−ω. The Gromov product 〈wi · wj〉 with respect to the basepoint ∅ is equal to \|w\| − l(g)/2, where w and g are as in Lemma 4.3. It follows that all the accumulation points of {wn} in X−ω are asymptotically equivalent to . . . x2x1. We also have that the sequence {xnxn−1 . . . x1}n≥1 converges to the same point of the hyperbolic boundary as {wn}. So the map D : X−ω −→ ∂Σ(G, S) is surjective and the map F : JG −→ ∂Σ(G, S), satisfying the conditions of the theorem, is uniquely defined. Let A = {g ∈ G : l(g) ≤ 4} and for every n ∈ N define Un = {(w1v, w2s(v)) : w1, w2 ∈ X−ω, v ∈ Xn, s ∈ A} ⊂ X−ω × X−ω. By Ũn we denote the image of Un in JG × JG. It is easy to prove now that ∩n≥1Un is equal to the asymptotic equiv- alence relation on X−ω and that Ũn is the basis of neighborhoods of a uniform structure, defining the topology on JG. It follows from Lemma 4.3 that Vn−2 ⊆ D × D(Un) ⊆ Vn. This implies that map F is a homeomorphism. References [BG00] Laurent Bartholdi and Rostislav I. Grigorchuk, On the spectrum of Hecke type operators related to some fractal groups, Proceedings of the Steklov Institute of Mathematics 231 (2000), 5–45. [BGN02] Laurent Bartholdi, Rostislav I. Grigorchuk, and Volodymyr V. Nekra- shevych, From fractal groups to fractal sets, Fractals in Graz 2001, Analysis– Dynamics–Geometry–Stochastics (Peter Grabner and Wolfgang Woess, eds.), Trends in Mathematics, vol. 19, Birkhäuser, 2003. Jo ur na l A lg eb ra D is cr et e M at h.86 Hyperbolic spaces from self-similar group actions [Bou71] Nicolas Bourbaki, Éléments de mathématique. Topologie générale. Chapitres 1 à 4, Hermann, Paris, 1971. [CDP90] Michel Coornaert, Thomas Delzant, and Athanase Papadopoulos, Geometrie et theorie des groupes: Les groupes hyperboliques de Gromov, Lectures Notes in Mathematics, vol. 1441, Springer Verlag, 1990. [GH90] Étienne Ghys and Pierre de la Harpe, Sur les groupes hyperboliques d’après Mikhael Gromov, Progress in Mathematics, vol. 83, Birkhäuser Boston Inc., Boston, MA, 1990, Papers from the Swiss Seminar on Hyperbolic Group held in Bern, 1988. [GNS00] Rostislav I. Grigorchuk, Volodymyr V. Nekrashevich, and Vitalĭı I. Sushchanskii, Automata, dynamical systems and groups, Proceedings of the Steklov Institute of Mathematics 231 (2000), 128–203. [Gri00] Rostislav I. Grigorchuk, Just infinite branch groups, New horizons in pro-p groups (Aner Shalev, Marcus P. F. du Sautoy, and Dan Segal, eds.), Progress in Mathematics, vol. 184, Birkhäuser Verlag, Basel, etc., 2000, pp. 121–179. [Gro87] Mikhael Gromov, Hyperbolic groups, Essays in Group Theory (S. M. Ger- sten, ed.), M.S.R.I. Pub., no. 8, Springer, 1987, pp. 75–263. [Kai03] Vadim A. Kaimanovich, Random walks on Sierpiński graphs: hyperbolicity and stochastic homogenization, Fractals in Graz 2001, Analysis–Dynamics– Geometry–Stochastics (Peter Grabner and Wolfgang Woess, eds.), Trends in Mathematics, vol. 19, Birkhäuser, 2003. [Nek02a] Volodymyr V. Nekrashevych, Iterated monodromy groups, preprint, Geneva University, 2002. [Nek02b] , Limit spaces of self-similar group actions, preprint, Geneva Univer- sity, 2002. [Nek] , Virtual endomorphisms of groups, Algebra and Discrete Mathemat- ics, Number 1, (2002), 96–136. [Sid98] Said N. Sidki, Regular trees and their automorphisms, Monografias de Matematica, vol. 56, IMPA, Rio de Janeiro, 1998. [Sul85] Dennis Sullivan, Quasi-conformal homeomorphisms and dynamics I, solu- tion of the Fatou-Julia problem on wandering domains, Ann. Math. 122 (1985), 401–418. Contact information V. Nekrashevych Kyiv Taras Shevchenko University Mechanics and Mathematics Faculty Algebra Department Volodymyrska, 60 Kyiv, Ukraine, 01033 E-Mail: nazaruk@ukrpack.net Received by the editors: 30.01.2003.

Hyperbolic spaces from self-similar group actions

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