Relations Between Various Boundaries of Relatively Hyperbolic Groups

Abstract

Communicated by O. KharlampovichSuppose a group G is relatively hyperbolic with respect to a collection P of its subgroups and also acts properly, cocompactly on a CAT(0) (or δ-hyperbolic) space X. The relatively hyperbolic structure provides a relative boundary ∂(G, P). The CAT(0) structure provides a different boundary at infinity ∂X. In this paper, we examine the connection between these two spaces at infinity. In particular, we show that ∂(G, P) is G-equivariantly homeomorphic to the space obtained from ∂X by identifying the peripheral limit points of the same type.Lemma 5.1. (Y gP ) gP∈Π is a quasidense, locally finite, bounded penetration and uniform neighborhood quasiconvex collection of subsets in X.Proof. It is obvious since Γ(G, S) and X are quasi-isometric under Φ, the collection (gP ) gP∈Π is quasidense, locally finite, bounded penetration and uniformly neighborhood quasiconvex in Γ(G, S), and (Y gP ) gP ∈Π is the image of (gP ) gP∈Π under the quasi-isometric map Φ. Lemma 5.2. ∂Y gP is non-empty if and only if gP is infinite and ∂YProof. The first statement is obvious and the second statement is implied by the bounded penetration property of (Y gP ) gP ∈Π .From the result of Lemma 5.2, the following concept is an equivalent relation. Int. J. Algebra Comput. 2013.23:1551-1572. Downloaded from www.worldscientific.com by UNIV OF ILLINOIS AT CHICAGO on 02/02/15. For personal use only. Relations Between Various Boundaries of Relatively Hyperbolic Groups 1563Definition 5.3. Two peripheral limit points are said to be of the same type if they both lie in ∂Y gP for some peripheral left coset gP .Lemma 5.4. If x and y are two peripheral limit points of the same type and g is any group element in G, then gx and gy are also two peripheral limit points of the same type.The proof for this lemma is obvious and we leave it to the reader.Remark 5.5. From the result of Lemma 5.4, we see that the group G acts on the space obtained from ∂X by identifying all peripheral limit points of the same type.Lemma 5.6. There are constants > 0, r > 0 such that the following holds. If α is a geodesic in X, then there is an -quasigeodesic c in Γ(G, S) such that the Hausdorff distance between Φ(c) and α is at most r. Moreover, if α is a geodesic segment with two endpoints Φ(g) and Φ(h), where g, h ∈ G, then c could be chosen with endpoints g and h. If α is a geodesic ray with initial point Φ(g), where g ∈ G, then c could be chosen with initial point g.

Boundary classification and two-ended splittings of groups with isolated flats

“…The strong connection between and the boundary is given by Hung Cong Tran . For spaces with isolated flats Tran's result implies that is the quotient space obtained from by identifying points which are in the boundary of the same flat.…”

Cannon–Thurston maps for $${{\,\textrm{CAT}\,}}(0)$$ groups with isolated flats

“…Bowditch [8] showed the pair (G, P) has a well-defined boundary, now called the Bowditch boundary and denoted ∂(G, P). The CAT(0) visual boundary contains spheres which arise as the visual boundary of flats, while these spheres are collapsed to points in the Bowditch boundary; see work of Tran [57]. Pal [48,Theorem 3.11] proved the existence of the Cannon-Thurston map for H G with respect to the Bowditch boundaries in the case that the group H is hyperbolic relative to a nontrivial subgroup H 1 , the group G is hyperbolic relative to N G (H 1 ) and weakly hyperbolic relative to H 1 , and G preserves cusps; see also [7,42].…”

“…Let be an axis for g. The endpoints α + , α − of the line lie in the boundary of the half-flat F 0 in ∂ X , and hence lie in the boundary of the flat F in ∂ X . By work of Tran [57,Main Theorem], there is a G-equivariant continuous map π : ∂ X → ∂(G, P), where ∂(G, P) denotes the Bowditch boundary for G defined with respect to the peripheral structure given by Hruska-Kleiner in Theorem 2.10. Moreover, there exists a parabolic fixed point z ∈ ∂(G, P) so that π(∂ F) = z.…”

“…As in the proof of Lemma 2.11, there is a continuous G-equivariant map π : ∂ X → ∂(G, P). Since an axis for b is contained in the flats F and F , there exists a parabolic point z ∈ ∂(G, P) so that π(F) = π(F ) = z and the elements a, a , and b stabilize z by[57, Main Theorem]…”

Boundaries of Dehn fillings

“…If the peripheral subgroups P of a relatively hyperbolic group pair (G, P) are themselves hyperbolic, then so is G, and it therefore acts as a uniform convergence group on its Gromov boundary ∂G. (The relationship between these boundaries is explained in [Tra13], see also [Ger12,GP13,MOY12,Man15].) In [Osi07,GM08] (cf.…”