G. Christopher Hruska

Relative hyperbolicity and relative quasiconvexity for countable groups

We lay the foundations for the study of relatively quasiconvex subgroups of relatively hyperbolic groups. These foundations require that we first work out a coherent theory of countable relatively hyperbolic groups (not necessarily finitely generated). We prove the equivalence of Gromov, Osin and Bowditch’s definitions of relative hyperbolicity for countable groups. We then give several equivalent definitions of relatively quasiconvex subgroups in terms of various natural geometries on a relatively hyperbolic group. We show that each relatively quasiconvex subgroup is itself relatively hyperbolic, and that the intersection of two relatively quasiconvex subgroups is again relatively quasiconvex. In the finitely generated case, we prove that every undistorted subgroup is relatively quasiconvex, and we compute the distortion of a finitely generated relatively quasiconvex subgroup. 20F65, 20F67

Hadamard spaces with isolated flats

We explore the geometry of nonpositively curved spaces with isolated flats, and its consequences for groups that act properly discontinuously, cocompactly, and isometrically on such spaces. We prove that the geometric boundary of the space is an invariant of the group up to equivariant homeomorphism. We also prove that any such group is relatively hyperbolic, biautomatic, and satisfies the Tits Alternative. The main step in establishing these results is a characterization of spaces with isolated flats as relatively hyperbolic with respect to flats. Finally we show that a CAT(0) space has isolated flats if and only if its Tits boundary is a disjoint union of isolated points and standard Euclidean spheres. In an appendix written jointly with Hindawi, we extend many of the results of this article to a more general setting in which the isolated subspaces are not required to be flats. AMS Classification numbers Primary: 20F67 Secondary: 20F69

Packing subgroups in relatively hyperbolic groups

Our main result establishes the bounded packing of relatively quasiconvex subgroups of a relatively hyperbolic group, under mild hypotheses. As an application, we prove that relatively quasiconvex subgroups have finite height and width, properties that strongly restrict the way families of distinct conjugates of the subgroup can intersect. We prove that an infinite, nonparabolic relatively quasiconvex subgroup of a relatively hyperbolic group has finite index in its commensurator. We also prove a virtual malnormality theorem for separable, relatively quasiconvex subgroups, which is new even in the word hyperbolic case. 20F65; 20F67, 20F69

Towers, ladders and the B. B. Newman Spelling Theorem

The Spelling Theorem of B. B. Newman states that for a one- relator group h a1; : : : j W n i, any nontrivial word which represents the identity must contain a (cyclic) subword of W n longer than W n 1 . We provide a new proof of the Spelling Theorem using towers of 2-complexes. We also give a geometric classication of reduced disc diagrams in one-relator groups with torsion. Either the disc diagram has three 2-cells which lie almost entirely along the boundary, or the disc diagram looks like a ladder. We use this ladder theorem to prove that a large class of one-relator groups with torsion are locally quasiconvex.

Commensurability invariants for nonuniform tree lattices

We study nonuniform lattices in the automorphism groupG of a locally finite simplicial treeX. In particular, we are interested in classifying lattices up to commensurability inG. We introduce two new commensurability invariants:quotient growth, which measures the growth of the noncompact quotient of the lattice; andstabilizer growth, which measures the growth of the orders of finite stabilizers in a fundamental domain as a function of distance from a fixed basepoint. WhenX is the biregular treeXm,n, we construct lattices realizing all triples of covolume, quotient growth, and stabilizer growth satisfying some mild conditions. In particular, for each positive real numberν we construct uncountably many noncommensurable lattices with covolumeν.

Hyperplane arrangements in negatively curved manifolds and relative hyperbolicity

We show that certain aspherical manifolds arising from hyperplane arrangements in negatively curved manifolds have relatively hyperbolic fundamental group.