My research is in the areas of geometric group theory and low-dimensional topology. In particular, I am interested in the relationship between negative curvature and the algebra, topology, geometry, analysis, and logic of groups. Most of my work has focused on the classes of hyperbolic, relatively hyperbolic and acylindrically hyperbolic groups, including many fundamental groups of 3-manifolds, mapping class groups of surfaces, right-angled Artin groups, and Out(Fn). I also use the properties of these groups to study related graphs, surfaces, and 3–manifolds. Some specific topics which I have thought or written about include:

Generalizations and applications of small cancellation theory.
Conjugacy growth.
Bounded cohomology and stable commutator length.
Asymptotic cones and quasi-isometric rigidity.
Highly transitive group actions.
Equations in groups, limit groups and actions on R–trees.
Random walks on groups.

 

All of my papers are available on arXiv.