Daniel T. Wise

A boundary criterion for cubulation

We give a criterion in terms of the boundary for the existence of a proper cocompact action of a word-hyperbolic group on a

Packing subgroups in relatively hyperbolic groups

{\rm CAT}(0)

FANS AND LADDERS IN SMALL CANCELLATION THEORY

cube complex. We describe applications towards lattices and hyperbolic 3-manifold groups. In particular, by combining the theory of special cube complexes, the surface subgroup result of Kahn-Markovic, and Agols criterion, we find that every subgroup separable closed hyperbolic 3-manifold is virtually fibered.

Hyperplane sections in arithmetic hyperbolic manifolds

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

Cubulating random groups at density less than 1/6

This paper provides a strengthening of the theorems of small cancellation theory. It is proven that disc diagrams contain ‘fans’ of consecutive 2-cells along their boundaries. The size of these fans is linked to the strength of the the small cancellation conditions satisfied by the diagram. A classification result is proven for disc diagrams satisfying small cancellation conditions. Any disc diagram either contains three fans along its boundary, or it is a ladder, or it is a wheel. Similar statements are proven for annular diagrams.

Finiteness properties of cubulated groups

In this paper, we prove that the homology groups of immersed totally geodesic hypersurfaces of compact arithmetic hyperbolic manifolds virtually inject in the homology group of the

Subgroup separability, knot groups and graph manifolds

We prove that random groups at density less than 1/6 act freely and cocompactly on CAT(0) cube complexes, and that random groups at density less than 1/5 have codimension-1 subgroups. In particular, Property (T ) fails to hold at density less than 1/5.

The Tits Alternative for Cat(0) Cubical Complexes

We give a generalized and self-contained account of Haglund–Paulin’s wallspaces and Sageev’s construction of the CAT(0) cube complex dual to a wallspace. We examine criteria on a wallspace leading to finiteness properties of its dual cube complex. Our discussion is aimed at readers wishing to apply these methods to produce actions of groups on cube complexes and understand their nature. We develop the wallspace ideas in a level of generality that facilitates their application. Our main result describes the structure of dual cube complexes arising from relatively hyperbolic groups. Let

Cubulating hyperbolic free-by-cyclic groups: the general case

H_1,\ldots, H_s

Separability of embedded surfaces in 3-manifolds

be relatively quasiconvex codimension-1 subgroups of a group