Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 21, 1996, 133–150

NEGATIVELY CURVED GROUPS AND THE CONVERGENCE PROPERTY II:

TRANSITIVITY IN NEGATIVELY CURVED GROUPS

Eric M. Freden

Utah State University, Uintah Basin Education Center 987 E Lagoon Roosevelt UT 84066, U.S.A.; freden@math.byu.edu

Abstract. Let G be a negatively curved group. This paper continues the classification of limit points of G that began in part I. A probability measure is constructed on the space at infinity and with respect to this measure almost every point at infinity is shown to be line transitive.

0. Introduction

Let M^{n} be a closed riemannian n-manifold with all sectional curvatures −1. Then π_{1}(M^{n}) is a discrete Möbius group acting properly discontinuously on the universal cover H^{n} and conformally on the boundary S^{n−1} . Since the action of π_{1}(M^{n}) is cocompact, every x ∈ S^{n−1} is a point of approximation (conical limit point) for π_{1}(M^{n}) (see 2.4.9 of [Nic] for instance). Furthermore, Γ (any Cayley graph for π_{1}(M^{n})) is quasi-isometric to H^{n} (see [Ca]) and so ∂Γ, the boundary at infinity of Γ, is homeomorphic to S^{n−1} . Therefore every x ∈ ∂Γ is a point of approximation.

Weakening the above hypotheses, assume M^{n} is any hyperbolic manifold. Let x ∈ S^{n−1} and suppose L ⊂ H^{n} is an oriented hyperbolic line with x as one endpoint. If for any b ∈ H^{n} there exists a sequence of distinct deck transformations {g_{n}} ⊂ G = π_{1}(M^{n}) such that the images g_{n}L come arbitrarily close to b, then x is called point transitive. If for any oriented hyperbolic line L′ there are distinct g_{n} such that g_{n}L → L′ preserving orientation, then x is line transitive or a Myrberg point. Clearly, every Myrberg point is also point transitive, but the converse is false in general (see [Sh]). Myrberg [M] first showed that for n = 2 the set of line transitive points has full Lebesgue measure in the boundary at infinity. More recently, Tukia has proved that in all dimensions the collection of conical limit points that are not Myrberg points is a nullset for any conformal G measure [T].

Let G be any negatively curved (Gromov hyperbolic) group with Cayley graph Γ and space at infinity ∂Γ. In I.3.7, the author showed that every x ∈ ∂Γ is a point of approximation. In this paper, almost every x ∈ ∂Γ is shown to be line transitive. The strategy is based on an old idea due to Artin. In [A], Artin showed that the line transitive points for the modular group SL(2, Z) are exactly those real numbers whose continued fraction expansion contains each finite sequence of integers. The continued fraction of a real number ξ can be obtained as the cutting sequence of a geodesic ray R ⊂ H^{n} tending to ξ (see 5.4 of [Se]). In the language of geometric group theory, a cutting sequence for R corresponds to reading the label of an equivalent geodesic ray R′ in the Cayley graph for SL(2, Z). Since SL(2, Z) ≃ Z_{2} ∗ Z_{3} is nearly a free group, cutting sequences and labels of geodesic rays are very similar [Se].

1991 Mathematics Subject Classification: Primary 20F32; Secondary 57S05.

This research was supported in part by the Geometry Center, University of Minnesota, an STC funded by the NSF, DOE, and Minnesota Technology, Inc.; and by an NSF research grant; also by Brigham Young University and Utah State University.

The concept of a geodesic ray containing each finite geodesic subsegment is somewhat similar to the idea of a normal number. Recall that a real number η is normal to base r if each block B_{k} of k digits occurs with frequency 1/r^{k}. It is well known that almost all real numbers are normal to every base (Chapter 8 [Niv] or 9.3–9.13 in [HW]). Hedlund and Morse generalized a similar concept to strings of symbols in their paper [HM]. Given a finite set of generating symbols and certain concatenation rules they considered infinite sequences containing a copy of every possible finite string of symbols. Such sequences were labelled transitive. Yet a third related subject concerns Markov chains, see [Fel]. A state (symbol, outcome, event, etc.) is persistent if with probability one, that state will recur (infinitely often) within the chain. This paper links all of the above.

Example 0.1. Suppose F = ⟨a, b⟩ is the free group of rank 2. Embed the associated Cayley graph in H^{2} . Each geodesic ray represents a unique point at infinity. Let w be a freely reduced word of length k in the generators. Emulating the counting estimate of [Niv], consider the ratio of the number of rays (from 0) of length nk containing w to the total number of rays of length nk. As n → ∞ this ratio tends to 1. The conclusion is that with respect to a certain natural (Cantor) measure, almost every ray contains w, and in fact infinitely often. By a transitive ray, I mean a geodesic ray from 0 that contains every finite geodesic word as a subpath. Theorem 1.4 below shows that any point represented by such a ray is line transitive (the proof of Theorem 1.4 in the case of a free group is much simpler than what I have written). The set of all geodesic segments form a Markov chain with four states: a, b, a^{−1}, b^{−1} . A given state can be succeeded by any state other than its inverse, with probability \frac{1}{3} . In the language of [Fel], each state is aperiodic, persistent, and has finite mean recurrence time, i.e., all states are ergodic.

In the case of a generic negatively curved G, relators (of perhaps arbitrarily long length) destroy the Markov aspect and vastly complicate the counting process. Showing that most points at infinity can be represented by an actual geodesic transitive ray seems to be difficult in the general case; in fact it may not be true! The difficulty is avoided by considering quasigeodesic rays.

The reader is expected to be familiar with [Fr], the first installment of “Negatively curved groups have the convergence property”. Results from that text will be prefaced by the the roman numeral I. Section 1 deals with definitions and preparatory material. Section 2 introduces the key simplification that reduces everything to symbolic dynamics. In Section 3 a measure on ∂Γ and the main theorems are established. I thank Jim Cannon for the idea behind Section 2.

1. Preliminaries

Throughout the remainder of the text, let G be a group with a fixed finite generating set such that the inverse of each generator is also a generator. (I do not allow the neutral element of G to be in the generating set.) Let Γ be the Cayley graph of G with respect to the given generators. Assume that Γ has thin triangle constant δ, boundary at infinity ∂Γ, and that G is non-elementary. Rays, halfspaces, etc. were defined in I.1.1–I.1.11. The ball with center a ∈ Γ and radius ρ > 0 is indicated by B(a, ρ). For ε > 0 and ray R, the neighborhood {a ∈ Γ : d(a, R) < ε} occurs so often that I will refer to it as the ε corridor about R.

Exercise 1.1. Let ε > 0. If R: [0, ∞) → Γ is a ray and r′, r ∈ (0, ∞) with r′ ≥ r + 2ε, then B(r′, ε) ⊂ H(R, r).

Definition 1.2. Suppose H is a non-elementary convergence group of the first kind (every point of X a limit point) acting on a compact Hausdorff space X. A point x ∈ X is said to be line transitive or a Myrberg point if given any two distinct points u, v ∈ X there exists a sequence of group elements {h_{n}} such that

h_{n}(x) → u and h_{n}(y) → v for all y ≠ x.

By passing to a subsequence we may suppose that the convergence is locally uniform away from x. If G is a non-elementary negatively curved group acting properly discontinously, cocompactly, and by isometry on a proper geodesic space Y , then G acts as a convergence group of the first kind on X = ∂Y and the above definition is equivalent to the following. If L is any geodesic line with one endpoint at a Myrberg point x and u ≠ v are points at infinity, then there exist {g_{n}} such that g_{n}L converges to a geodesic joining u and v with g_{n}(x) → u. This explains the term “line transitive”. (If Y is classical hyperbolic m-space, such a geodesic is unique, whereas in a Cayley graph there are in general many geodesics joining a pair of points at infinity.) Definition 1.2 requires no domain of discontinuity for G, merely a non-elementary limit set. (I have been unable to find an analogous generalization for point transitivity.)

Figure 1.1

Lemma 1.3. Let R, S ⊂ Γ be rays, r ∈ R, and s_{1} ∈ S ∩ H^{−}(R, r). If s_{2} ∈ S ∩ H(R, r + 14δ) with s_{2} > s_{1} , then S^{+} ⊂ H(R, r) for the sub-ray S^{+} = S([s_{2}, ∞)) of S.

Proof. Suppose, by way of contradiction, that S exits H(R, r) after s_{2} (see Figure 1.1). Let s_{3} be a point in S^{+} ∩ H^{−}(R, r). Then \overline{s_{1}s_{3}} is a geodesic segment with both endpoints in H^{−}(R, r) with s_{2} ∈ \overline{s_{1}s_{3}} . By Lemma I.2.4, the halfspace H^{−}(R, r) is quasiconvex(2δ) so d(s_{2}, H^{−}(R, r)) < 2δ. On the other hand, Lemma I.2.5 implies that d(s_{2}, H^{−}(R, r) is greater than (14 − 12)δ = 2δ, a contradiction.

Theorem 1.4. Suppose x ∈ ∂Γ is represented by a ray R from 0, and that for every finite geodesic word, R contains a directed subpath labelled with the given word. Then x is a Myrberg point.

Proof. Let u ≠ v ∈ ∂Γ be represented by rays S, T respectively. Denote the vertices (group elements) of S, T tending from 0 by s_{1}, s_{2}, … and t_{1}, t_{2}, …. Given n > 0, the literal edge path t_{n}^{−1}s_{n} is almost surely not geodesic; I will abuse notation and denote by \overline{t_{n}^{−1}s_{n}} any geodesic path between the vertices t_{n} and s_{n}. Using an easy thin triangles argument (see 7.5 of [G] or p. 19 of [CDP]) there is a bound M > 0 independent of n and an intermediate point a_{n} of t_{n}^{−1}s_{n} such that d(a_{n}, 0) ≤ M.

By hypothesis, the ray R contains \overline{t_{n}^{−1}s_{n}} as a segment, meaning R = r_{n}\overline{t_{n}^{−1}s_{n}} ⋯ for some initial geodesic segment r_{n}. For each n > 0 define g_{n}: Γ → Γ as left multiplication by t_{n}r_{n}^{−1} . Then g_{n}(0) = t_{n}r_{n}^{−1} , g_{n}(r_{n}) = t_{n}, and g_{n}(r_{n}t_{n}^{−1}s_{n}) = s_{n} (refer to Figure 1.2). Pick any s ∈ S, t ∈ T so that both halfspaces H(S, s), H(T, t) are disjoint from the ball B(0, M). Choose n so large that s_{n} ≥ s + 22δ and t_{n} ≥ t + 14δ. Evidently g_{n}(R) is a ray passing through t_{n}, a_{n}, and s_{n}, in that order, where a_{n} is an intermediate point of the subsegment \overline{t_{n}^{−1}s_{n}} such that d(a_{n}, 0) ≤ M. Note that a_{n} ∈ H^{−}(T, t) and t_{n} ∈ H(T, t + 14δ). From the proof of Lemma 1.3, g_{n}(0) = t_{n}r_{n}^{−1} ∈ H(T, t), and by the same argument, the tail of g_{n}(R) starting at s_{n} lies entirely within H(S, s + 8δ). Therefore the endpoint g_{n}(x) ∈ \overline{D(S, s + 8δ)} ⊂ D(S, s), the last inclusion following from 1.12 of [Sw]. This shows that g_{n}(0) → v and g_{n}(x) → u as n → ∞. Use the convergence property to find a subsequence so that g_{j} → v locally uniformly on \overline{Γ} \ \{x\}.

Figure 1.2

2. Symbols, words, and quasigeodesics

For each positive integer p, let 𝒟_{p} denote the cover of ∂Γ by open combinatorial disks {D(R, p) : R is a ray from 0}. Lemma 1.14 of [Sw] shows that if R, S are rays from 0 with d(R(p), S(p)) > 18δ then the corresponding disks D(R, p), D(S, p) are disjoint. Furthermore, 1.15 of [Sw] says that D(R, p) is uniquely determined by the segment R([p − 4δ, p + 4δ]) ⊂ R. These two results imply that there is some P > 0 such that for any fixed p > 4δ, the closure of a given D ∈ 𝒟_{p} intersects at most P other closed disks in the collection 𝒟_{p}. Let {𝒟_{p}}_{p>4δ}^{∞} be such a sequence of disk covers of ∂Γ. Fix p > 7δ and for each D(R, p) ∈ 𝒟_{p}, let O(R, p) denote an open subset of Γ̄ = Γ ∪ ∂Γ such that

D(R, p) ⊂ O(R, p) ⊂ [H(R, p) ∪ D(R, p)].

Using any global metric on Γ̄ we may suppose that O(R, p) is so close (pointwise) to D(R, p) that

D(R, p) ∩ D(S, p) = ∅ implies O(R, p) ∩ O(S, p) = ∅.

Set 𝒪_{p} = {O(R, p)}. We may assume that p is so large that there are at least 3P members of 𝒪_{p}.

Lemma 2.1. The distance from any O(R, p) to 0 is at least 4δ.

Figure 2.1

Proof. It is sufficient to show that the distance from H(R, p) to the origin is at least 4δ (see Figure 2.1). Suppose that a ∈ H(R, p) is within 4δ of 0. Then a projects to some point b ∈ R[ p, ∞). Consider the triangle with vertices 0, a, b. Since a is in the halfspace, the length of \overline{ab} is at most 4δ =d(a, 0). A simple thin triangles exercise (made explicit in 1.3 of [Sw]) shows that

7δ ≤ d(0, b) ≤ d(0, a) + 2δ ≤ 4δ + 2δ = 6δ

which is a contradiction.

Pick q > p so large that the complement of B(0, q) ⊂ Γ is contained in ∪𝒪_{p}. Assign a distinct symbol to each element in G of length q. Let 𝒜_{q} be the alphabet of symbols having (reduced word) length q. Assign to each s ∈ 𝒜_{q} a shortest representative word in G, or equivalently, a geodesic edge path from 0 to s in the Cayley graph. Define a formal concatenation of symbols as follows. Let s, t ∈ 𝒜_{q}. The product s· t is admissible if s^{−1} and t, considered as edge paths from 0 in the Cayley graph, terminate in O(R, p), O(S, p) respectively, with O(R, p) ∩ O(S, p) = ∅. A word in the alphabet 𝒜_{q} is admissible if each subword of length 2 is admissible. Let 𝒲_{q} be the set of all admissible words in the alphabet 𝒜_{q}.

Definition 2.2 (see 11.15 of [Ca]). Let N > 0, a, b, c ∈ Γ, and suppose \overline{ab}, \overline{bc} are geodesic segments. Define the N -deviation of the broken geodesic segment \overline{ab} ∪ \overline{bc} at b by

dev_{N}( \overline{ab} ∪ \overline{bc}) = sup\{d(x, b), d(y, b) : d(x, y) ≤ N, x ∈ \overline{ab}, y ∈ \overline{bd}\}.

The deviation is a measure of the angle at which \overline{ab} and \overline{bc} meet. If \overline{ab} ∪ \overline{bc} is geodesic, then the N -deviation of their union is N.

Deviation Lemma 2.3. Given the collection 𝒪_{p}, there exists d > 0 such that for all sufficiently large q and for all s, t ∈ 𝒜_{q} with s· t admissible, it is true that the 3δ -deviation of s^{−1} ∪ t is less than d.

Proof. Suppose the conclusion is false. Then there exists an unbounded increasing sequence of positive integers q_{n} with corresponding alphabets and word sets 𝒜_{q_{n}}, 𝒲_{q_{n}}, respectively, and symbols s_{q_{n}}, t_{q_{n}} ∈ 𝒜_{q_{n}} such that s_{q_{n}}· t_{q_{n}} is admissible and

dev_{3δ}(s_{q_{n}}^{−1} ∪ t_{q_{n}}) ≥ n.

Figure 2.2

By passing to subsequences we may assume that s_{q_{n}}^{−1} , t_{q_{n}} are represented by edge paths that are initial substrings of the edge paths representing s_{q_{n+1}}^{−1} , t_{q_{n+1}} , respectively, for all n. Thus the increasing unions

∪s_{q_{n}}^{−1} and ∪ t_{q_{n}}

correspond to geodesic rays S, T in the Cayley graph. Given a fixed q_{n} > 0 there are points s ∈ S and t ∈ T with d(s, t) ≤ 3δ and (without loss of generality) d(0, s) ≥ n (refer to Figure 2.2). Using the fact that triangle(s0t) is δ -thin it is evident that \overline{0s} is in the 4δ corridor about T. Furthermore, d(0, t) is at least n − 3δ. Since the above argument works for arbitrarily large n it follows that S is in the 4δ corridor about T, T is in the 4δ corridor about S, and thus S and T are in fact equivalent rays. Evidently S and T end in the same D(R, p) ⊂ O(R, p). This contradicts the fact that each s_{q_{n}}· t_{q_{n}} is admissible.

Given d from the deviation lemma fix q > max{p, 2d + 12δ} so large that Γ \ B(0, q) is contained in the union of all the O(R, p). Let 𝒜 = 𝒜_{q}, 𝒲 = 𝒲_{q}, 𝒪 = 𝒪_{p} denote the corresponding alphabet, set of admissible words, and collection of open sets, respectively.

Recall that a path Q in Γ is quasigeodesic(K) if each subpath Q′ = \overline{ab} of Q having length at least K satisfies

d(a, b) ≥ (\frac{1}{K} ) length(Q′).

Corollary 2.4. There exists K > 0 such that each w ∈ 𝒲 is quasigeodesic(K).

Proof. By 11.16 of [Ca] and the deviation lemma, each w ∈ 𝒲 is quasigeodesic(K) with K = q − (2d + 12δ).

Furthermore, if a, b denote the endpoints of w as a path in Γ and \overline{ab} is any geodesic segment from a to b, then there exists some N > 0 depending only on K and Γ such that w lies in the N corridor about \overline{ab} (see 11.20 of [Ca]).

Definition 2.5. Let 𝒲_{∞} be the set of all infinite sequences of symbols a_{0}· a_{1}· a_{2} ⋯ such that a_{i−1}· a_{i} is admissible for all i ≥ 1. A cylinder is a tree; namely a set of the form

𝒮 = {a_{0}· a_{1} ⋯ a_{i}· a_{i+1} ⋯ ∈ 𝒲_{∞} : a_{0}· a_{1} ⋯ a_{i} is fixed }.

The initial fixed edges a_{0}· a_{1} ⋯ a_{i} form the stem. Unless otherwise noted, all cylinders are to be regarded as having stems based at the origin. The other end of the stem is the branch vertex. The level of a (non-stem) vertex v in a cylinder is the number of edges between the branch vertex and v.

Other authors define a more general cylinder, see 2.4.1 of [K]. It is not hard to see that the cylinders form a base for a topology that makes 𝒲_{∞} a Cantor set. (The reader should realize at this point that 𝒲_{∞} is an abstract topological space distinct from the Cayley graph of G. There is an obvious quotient map from 𝒲_{∞} onto ∂Γ that will be used in the next section.) The usual middle thirds Cantor set can be visualized as the ends of a regularly branching tree while the tree for 𝒲_{∞} ostensibly branches in an irregular fashion (see Figure 2.3).

Corollary 2.4 implies that each a_{0}· a_{1}· a_{2} ⋯ ∈ 𝒲_{∞} is a quasi-ray (quasigeodesic ray) contained in the N corridor about some ray R. The converse is also true.

Approximation Lemma 2.6. Given any geodesic ray R ⊂ Γ from 0, there exists a quasi-ray W = a_{0}· a_{1}· a_{2} ⋯ ∈ 𝒲_{∞} such that R lies inside the δ corridor about W and W is in the δ corridor about R.

Figure 2.3

Proof. The quasi-ray W is defined inductively. For n = 0, pick v_{0} to be the unique vertex (group element) on R satisfying d(0, v_{0}) = q. There is a symbol a_{0} ∈ 𝒜 which, interpreted as an edge path from 0, terminates at v_{0} . Assume the string a_{0}· a_{1} ⋯ a_{n−1} has been constructed. Let v_{n} be the unique vertex on R satisfying d(0, v_{n}) = (n + 1)q. There is a symbol a_{n} ∈ 𝒜 such that the edge path a_{0}· a_{1} ⋯ a_{n−1}· a_{n} terminates at v_{n}.

Suppose, by way of contradiction, that a_{n−1}· a_{n} is not admissible. Then there exist O, O′ ∈ 𝒪 such that a_{n−1}^{−1}, a_{n} are vertices in O, O′ respectively, with O ∩ O′ ≠ ∅. Pick b ∈ O ∩ O′ and consider any geodesic segments connecting a_{n−1}^{−1}, a_{n} with b. Note that a_{n−1}^{−1} and a_{n} are joined by a translated segment of R passing through 0 (refer to Figure 2.4). By thinness of the triangle with vertices a_{n−1}^{−1} , b, and a_{n}, it follows that d(0, \overline{a_{n−1}^{−1}b} ∪ \overline{ba_{n}}) ≤ δ. Without loss of generality suppose that d(0, \overline{a_{n−1}^{−1}b}) ≤ δ.

Figure 2.4

By construction, O is contained in a halfspace H of the form H(R, p). Using I.2.4, the latter halfspace is quasiconvex(2δ), meaning that the entire segment \overline{a_{n−1}^{−1}b} stays within 2δ of H. Therefore the distance from the origin to the open set O is at most δ + 2δ = 3δ. This contradicts Lemma 2.1, therefore a_{n−1}· a_{n} is admissible.

The infinite word a_{0}· a_{1}· a_{2} ⋯ labels a geodesic ray in Γ which at levels 0, q, 2q, ⋯ coincides with the vertices 0, v_{0}, v_{1}, ⋯ of R. It is clear that the subpath labelled by a_{n} can stray no farther than \frac{1}{2}q from R. In fact, this bound can be improved to δ: Let v_{n−1}, v_{n} be the vertices on R corresponding to the endpoints of the edge path a_{n}. The geodesic digon with vertices v_{n−1}, v_{n} and edges from R and W is δ -thin.

Corollary 2.7. If w ≠ w′ ∈ 𝒲 , there is some a ∈ 𝒜 such that w· a· w′ ∈ 𝒲 .

Proof. Let s, t ∈ 𝒜 denote the last symbol in w and the first symbol in w′, respectively. By design, there are at most P inadmissible suffixes to s and P inadmissible prefixes for t, but at least 3P symbols in 𝒜 . Let a ∈ 𝒜 be any one of the (at least 3P − P − P = P) symbols such that s· a· t is admissible. Then w· a· w′ ∈ 𝒲 .

The hypotheses of Theorem 1.4 may be impossible to satisfy for arbitrary negatively curved groups. The weaker condition that x ∈ ∂Γ is represented by a quasi-ray R′ ∈ 𝒲_{∞} satisfying the property that R′ contains every finite admissible word is sufficient to guarantee that x is line transitive.

Theorem 2.8. Suppose x ∈ ∂Γ is represented by a quasi-ray R′ ∈ 𝒲_{∞} , and that every w ∈ 𝒲 occurs as a substring of R′. Then x is a Myrberg point.

Proof. The proof is modelled on Theorem 1.4. Quasigeodesics behave in the large very much like geodesics and only some of the constants need be changed.

Figure 2.5

Let R′ = ρ_{1}· ρ_{2}· ρ_{3} ⋯ ∈ 𝒲_{∞} be any transitive quasi-ray from 0 representing x in the N corridor about some equivalent geodesic ray R. Suppose u ≠ v ∈ ∂Γ, and let S, T be rays from 0 representing u, v respectively. Using the approximation Lemma 2.6 there are quasi-rays σ_{1}· σ_{2}· σ_{3} ⋯ and τ_{1}· τ_{2}· τ_{3} ⋯ in the δ corridors about S, T respectively.

By Corollary 2.7 there exist admissible quasigeodesic segments connecting each τ_{n} to σ_{n}, considered as vertices in Γ. I will abuse notation and refer to any such segment as \overline{τ_{n}^{−1}σ_{n}}. By hypothesis, the admissible quasi-ray ρ_{1}· ρ_{2}· ρ_{3} ⋯ contains \overline{τ_{n}^{−1}σ_{n}} as a subsegment, i.e., the quasi-ray R′ contains ρ_{k_{n}}· (τ_{n}^{−1}σ_{n}) as an initial segment for some ρ_{k_{n}} ∈ 𝒲 . For each n, define g_{n} as left multiplication by τ_{n}ρ_{k_{n}}^{−1} . Then

g_{n}(0) = τ_{n}ρ_{k_{n}}^{−1} and g_{n}(ρ_{k_{n}}(τ_{n}^{−1}σ_{n})) = σ_{n}.

Since R′ behaves very much like a geodesic, it is not hard to see that there is a constant M > 0 independent of n and a point α_{n} ∈ g_{n}(R′) such that d(α_{n}, 0) ≤ M for each n. Also there exist s_{n}, t_{n} in S, T respectively, such that d(s_{n}, σ_{n}) < δ and d(t_{n}, τ_{n}) < δ. Clearly s_{n} → u and t_{n} → v as n → ∞. Let s ∈ S and t ∈ T so that H(S, s) ∩ B(0, M + N) = ∅ = H(T, t) ∩ B(0, M + N). Pick n so large that t_{n} > t + 2N + 16δ and s_{n} > s + 2N + 24δ and choose r_{n} ∈ R within N of ρ_{k_{n}} = g_{n}^{−1}(τ_{n}) ∈ R′. Then g_{n}(r_{n}) ∈ g_{n}(R) is within N +δ of t_{n} ∈ T. Exercise 1.1 implies that g_{n}(r_{n}) ∈ B(t_{n}, N + δ) ⊂ H(T, t_{n} − 2(N + δ)) ⊂ H(t, t + 14δ), while an intermediate point of g_{n}(R) near α_{n} is inside B(0, M + N) ⊂ H^{−}(T, t). Lemma 1.3 says the image of R between 0 and ρ_{k_{n}} is inside H(T, t). Therefore g_{n}(0) → v.

Similar changes in constants show that a tail of g_{n}(R) ⊂ H(S, s). The details are left to the reader.

Although the word space 𝒲 is a groupoid rather than a group, it possesses an automatic structure in the sense that there exists a deterministic finite state automaton that decides if w· s ∈ 𝒲 for any s ∈ 𝒜 , w ∈ 𝒲 . In fact all the decision making data can be stored in a square (0, 1)-matrix M = (m_{ij}) of order |𝒜 |, the cardinality of the q -sphere in Γ. Enumerate 𝒜 = {s_{1}, s_{2}, …}. Set

m_{ij} = \{ \begin{matrix}1, & if the product s_{i}· s_{j} is admissible; \\ 0, & else. \end{matrix}

This construction makes 𝒲 a Markov chain, see pp. 2, 88 of [R] or Chapter XV of [Fel]. Corollary 2.7 implies that the matrix M is primitive of exponent 3 (i.e., each entry of M^{3} is strictly positive). Further links between graph and matrix theories are given in [B].

The approximation lemma implies that every point of ∂Γ is represented by at least one quasi-ray W ∈ 𝒲_{∞}. The boundary at infinity is complicated by the local interaction of relators in Γ. These difficulties can be reduced by considering the simpler structure of 𝒲_{∞}.

3. Measures on 𝒲_{∞} and ∂Γ

There is a natural Cantor measure ν on 𝒲_{∞} such that ν(𝒲_{∞}) = 1. To each a ∈ 𝒜 let C(a) denote the number of admissible suffixes for a. If 𝒮_{1} ⊂ 𝒲_{∞} is the cylinder of all quasi-rays with fixed initial symbol a_{1} , define ν(𝒮_{1}) = 1/|𝒜 | . Inductively, if 𝒮_{n} ⊂ 𝒲_{∞} is the set of quasi-rays whose first n entries consist of the fixed symbols a_{1} ⋯ a_{n} and 𝒮_{n+1} ⊂ 𝒮_{n} consists of those quasi-rays satisfying the additional condition that the n+1^{st} position contains the symbol a_{n+1} , define

ν(𝒮_{n+1}) = ν(𝒮_{n})\frac{1}{C(a_{n})} = (\frac{1}{|𝒜 |} )(\frac{1}{C(a_{1})}) ⋯ (\frac{1}{C(a_{n})}).

The cylinders of the type 𝒮_{n} are analogous to open intervals in the construction of Lebesgue measure on the unit interval (refer to Figure 2.3). For any Borel set ℰ ⊂ 𝒲_{∞} define ν(ℰ) as the infimum of ∑ ν(𝒮) where ∪𝒮 is any countable covering of ℰ by basic open cylinders of the above types. In this way ν becomes a non-atomic regular Borel measure on 𝒲_{∞} with total mass 1.

By construction of 𝒪, 𝒜 , 𝒲 there is a uniform upper bound of C for C(a). Suppose W = a_{1}· a_{2}· a_{3} ⋯ ∈ 𝒲_{∞}. For a given s ∈ 𝒜 , let α_{2n} be the probability that the first occurrence of s in W is s = a_{2n−1} or s = a_{2n}. Then the probability that s occurs as a symbol in W is ∑_{n=1}^{∞} α_{2n}. Define β_{2n} = ∑_{i=1}^{n} α_{2i}. The claim is that B = lim_{n→∞} β_{2n} is in fact 1.

Let 𝒲_{2n} be the set of all words in 𝒲 of symbolic length at most 2n. Let 𝒰_{2n} ⊂ 𝒲_{2n} be those words containing s, and let 𝒱_{2n} = 𝒲_{2n} \ 𝒰_{2n}. By definition, β_{2n} = ν(𝒰_{2n}).

Suppose v ∈ 𝒱_{2n}. By Corollary 2.7, there is some a ∈ 𝒜 such that v· a· s ∈ 𝒲 . Hence the measure of the set of quasi-rays beginning in v and having s as symbol 2n + 1 or 2n + 2 is at least 1/C^{2} times the measure of the cylinder set

𝒮 (v) = {W ∈ 𝒲_{∞} : W has v as initial segment}.

Consequently

α_{2n} ≥ \frac{1}{C^{2}} ν(𝒱_{2n}) = \frac{1}{C^{2}} (1 − β_{2n−2}) ≥ \frac{1}{C^{2}} (1 − B).

But ∑ α_{2n} ≤ ν(𝒲_{∞}) = 1, so that α_{2n} ↓ 0. Therefore 1 − B = 0. This proves

Lemma 3.1. Let s ∈ 𝒜 and suppose W ∈ 𝒲_{∞} is a quasi-ray. Then with probability 1, W contains s as a symbol.

Scholium 3.2. Suppose that s ∈ 𝒲 . Then with probability 1, W contains s as a subword. (The set of all W ∈ 𝒲_{∞} containing s has full ν measure.)

Proof. The argument given above need only be modified slightly. Let k be the length of s in the alphabet 𝒜 . Note that if w ∈ 𝒲 , then w· s is admissible as soon as w· s_{1} is admissible, where s_{1} is the first symbol of s. Write W = w_{1}· w_{2} ⋯ where each w_{n} ∈ 𝒲 consists of k + 1 symbols. Define α_{n} as the probability that s first appears in W as a substring of w_{n}. The probability that s occurs in at least one of the blocks w_{i} is ∑_{n=1}^{∞} α_{n}. Let β_{n} be the n^{th} partial sum. As in the previous argument, given a prefix w_{1}· w_{2} ⋯ w_{n−1} there is at least one admissible word w_{n−1}· a· s for some a ∈ 𝒜 . Thus

α_{n} ≥ \frac{1}{C^{k+1}} (1 − β_{n−1}).

If B = lim_{n→∞} β_{n} < 1, then just as in Lemma 3.1, α_{i} ≥ (1 − B)/C^{k+1} > 0 and ∑ α_{i} diverges, a contradiction.

In the language of symbolic dynamics, the measure ν makes 𝒲_{∞} an ergodic Markov process with respect to the shift map that sends a quasi-ray W ∈ 𝒲_{∞} to the same ray less its first symbol. All the relevant information for ν can be encoded in a symmetric doubly-stochastic matrix Π = (π_{ij}) defined by

π_{ij} = \{ \begin{matrix}1/C(a_{i}), & if m_{ij} = 1; \\ 0, & else, \end{matrix}

where M = (m_{ij}) is the matrix of admissibility discussed at the conclusion of Section 2.

Let f: 𝒲_{∞} → ∂Γ be the quotient map that identifies each quasi-ray to its endpoint at infinity. Then f is a proper continuous surjection between compact metric spaces. The map f induces a measure μ on ∂Γ by

μ(E) = ν(f^{−1}(E)) for each E ⊂ ∂Γ.

From 2.2.17 of [Fed], μ is a regular Borel (Radon) measure on ∂Γ. The next lemmas show that μ has no atoms.

Lemma 3.3. Let 𝒮 ⊂ 𝒲_{∞} be a cylinder and m_{0} a positive integer. Divide 𝒮 into levels of length m_{0} . Throw out one branch (and its descendants) from each subcylinder based at levels 0, m_{0}, 2m_{0}, 3m_{0}, …. The remaining set ℰ satisfies ν(ℰ) = 0.

Proof. First consider the case when m_{0} = 1 and 𝒮 is a regular tree of valence C with root at the origin, i.e., 𝒮 is the entire symbol space 𝒲_{∞}. It is easy to see (Figure 3.1) that the measure of the complement of ℰ is the sum of a geometric series.

ν(𝒮 \ ℰ) = \frac{1}{C} + \frac{C − 1}{C^{2}} + \frac{(C − 1)^{2}}{C^{3}} + ⋯ = 1

Thus ν(ℰ) = 0.

Figure 3.1

For m_{0} > 1, consider 𝒮 as a regular tree of valence m_{0}C, where initial edges have been identified (see Figure 3.2). In terms of the measure ν, nothing has been changed:

ν(𝒮 \ ℰ) = \frac{1}{m_{0}C} + \frac{m_{0}C − 1}{(m_{0}C)^{2}} + \frac{(m_{0}C − 1)^{2}}{(m_{0}C)^{3}} + ⋯ = 1.

In the general case 𝒮 has an initial segment, branches irregularly, but has valence bounded by some C > 0. Omitting one branch at each m_{0} level results in a set ℰ having ν -measure less than or equal to the corresponding case of a regular tree of valence C.

Figure 3.2

Lemma 3.4. Let ξ ∈ ∂Γ, and suppose R ⊂ Γ is a geodesic ray representing ξ. There exists N > 0 such that any quasi-ray W representing ξ lies entirely within the N + 2δ corridor about R.

Proof. The remarks following Corollary 2.4 imply that there is some (globally defined) N > 0 such that any quasi-ray W representing ξ lies in the N corridor of some ray S. Clearly S represents ξ. By I.1.7 the ray S lies inside the 2δ corridor about R.

Lemma 3.5. Let ξ ∈ ∂Γ. There exists a positive integer m_{0} such that for any geodesic ray R ending at ξ and any cylinder 𝒮 whose branch vertex is within N + 2δ of R, there exists a branch of 𝒮 which at level m_{0} is farther than 2(N + δ) from R. (Translation: any cylinder whose branch point is in the N + 2δ corridor of a ray has a branch that exits that corridor shortly. See Figure 3.3.)

Sketch of proof. Following ideas in [CS], there is a global constant m_{1} such that non-equivalent geodesic rays diverge more than N + 2δ from each other (and continue to diverge) after traveling m_{1} units from a common vertex. There exists a similar constant m_{0} ≥ m_{1} for which the analogous result holds for inequivalent quasi-rays based at the same vertex (the details are left to the reader). By construction every cylinder contains non-equivalent quasi-rays (in fact the f -image of a cylinder contains some disk D(T, t) and thus has non-empty interior in ∂Γ).

Figure 3.3 ξ

Theorem 3.6. μ is non-atomic.

Proof. Let ξ ∈ ∂Γ be represented by a ray R. By Lemma 3.4, all quasirays mapping to ξ lie in the N + 2δ -corridor about R. Let {𝒮_{1}, 𝒮_{2}, 𝒮_{3}, …} be the collection of all cylinders containing f^{−1}(ξ). This set is countable since the totality of all cylinders is countable. Let 𝒮_{i} be such a cylinder. Evidently 𝒮_{i} has its branch vertex inside the N + 2δ corridor about R (else no quasi-ray of 𝒮_{i} can map to ξ by Lemma 3.4). Lemma 3.5 says that at level m_{0} there is a branch of 𝒮_{i} that exits the corridor, i.e., this branch is disjoint from f^{−1}(ξ). As in Lemma 3.3, let ℰ_{i} ⊂ 𝒮_{i} be the set of those branches that remain to (possibly) map to ξ. Lemma 3.3 says that ν(ℰ) = 0. Thus

0 ≤ μ(ξ) = ν(f^{−1}(ξ)) ≤ ∑_{i} ν(ℰ_{i}) = 0.

Main Theorem 3.7. The set of Myrberg points in ∂Γ has full μ measure.

Proof. Let ℰ be the set of all quasi-rays in 𝒲_{∞} that contain every finite subword. By Scholium 3.2, ν(ℰ) = 1, and ℰ ⊂ f^{−1}(f(ℰ)) is always true. Therefore

μ(f(ℰ)) = ν(f^{−1}(f(ℰ))) ≥ ν(ℰ) = 1.

It is sufficient to show that each x ∈ f(ℰ) is a Myrberg point. Theorem 2.8 does precisely this.

Comments and Conjectures 3.8. Michel Coornaert has recently defined a family of quasi-conformal measures for the boundary of a Gromov hyperbolic group [Co]. Hausdorff measure is one such example. I conjecture that μ defined above is a quasi-conformal measure. Negatively curved groups have no parabolic elements. Does an analog of Theorem 3.7 hold for arbitrary discrete isometry groups of negatively curved (Gromov hyperbolic) geodesic spaces? Myrberg points are defined topologically (Definition 1.2) whereas my methods are geometric. Does Theorem 3.7 depend on the geometry of the Cayley graph or is there a topological proof, in particular, does Theorem 3.7 hold for convergence groups of the first kind?

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Received 22 August 1994 Received in revised form 8 January 1996