We develop a theory of strongly quasiconvex subgroups of an arbitrary finitely generated group. Strong quasiconvexity generalizes quasiconvexity in hyperbolic groups and is preserved under quasiisometry. We show that strongly quasiconvex subgroups are also more reflexive of the ambient groups geometry than the stable subgroups defined by Durham-Taylor, while still having many analogous properties to those of quasiconvex subgroups of hyperbolic groups. We characterize strongly quasiconvex subgroups in terms of the lower relative divergence of ambient groups with respect to them.We also study strong quasiconvexity and stability in relatively hyperbolic groups, right-angled Coxeter groups, and right-angled Artin groups. We give complete descriptions of strong quasiconvexity and stability in relatively hyperbolic groups and we characterize strongly quasiconvex special subgroups and stable special subgroups of two dimensional right-angled Coxeter groups. In the case of right-angled Artin groups, we prove that two notions of strong quasiconvexity and stability are equivalent when the right-angled Artin group is one-ended and the subgroups have infinite index. We also characterize non-trivial strongly quasiconvex subgroups of infinite index (i.e. non-trivial stable subgroups) in right-angled Artin groups by quadratic lower relative divergence, expanding the work of Koberda-Mangahas-Taylor on characterizing purely loxodromic subgroups of right-angled Artin groups.

Hierarchically hyperbolic spaces (HHSs) are a large class of spaces that provide a common frame work for studying the mapping class group, right-angled Artin and Coxeter groups, and many 3-manifold groups. We investigate quasiconvex subsets in this class and characterize them in terms of their contracting properties, relative divergence, the coarse median structure, and the hierarchical structure itself. Along the way, we obtain new tools to study HHSs, including two new equivalent definitions of hierarchical quasiconvexiy and a version of the bounded geodesic image property for quasiconvex subsets. Utilizing our characterization, we prove that the hyperbolically embedded subgroups of hierarchically hyperbolic groups are precisely those which are almost malnormal and quasiconvex, producing a new result in the case of the mapping class group. We also apply our characterization to study quasiconvex subsets in several specific examples of HHSs. We show that while many commonly studied HHSs have the property that that every quasiconvex subset is either hyperbolic or coarsely covers the entire space, right-angled Coxeter groups exhibit a wide variety of quasiconvex subsets.

We determine which amalgamated products of surface groups identified over multiples of simple closed curves are not fundamental groups of 3-manifolds. We prove each surface amalgam considered is virtually the fundamental group of a 3-manifold. We prove that each such surface group amalgam is abstractly commensurable to a right-angled Coxeter group from a related family. In an appendix, we determine the quasi-isometry classes among these surface amalgams and their related right-angled Coxeter groups.

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.

We show the mapping class group, $${{\,\mathrm{CAT}\,}}(0)$$ CAT ( 0 ) groups, the fundamental groups of closed 3-manifolds, and certain relatively hyperbolic groups have a local-to-global property for Morse quasi-geodesics. This allows us to generalize combination theorems of Gitik for quasiconvex subgroups of hyperbolic groups to the stable subgroups of these groups. In the case of the mapping class group, this gives combination theorems for convex cocompact subgroups. We show a number of additional consequences of this local-to-global property, including a Cartan–Hadamard type theorem for detecting hyperbolicity locally and discreteness of translation length of conjugacy classes of Morse elements with a fixed gauge. To prove the relatively hyperbolic case, we develop a theory of deep points for local quasi-geodesics in relatively hyperbolic spaces, extending work of Hruska.