## Josh Barnard's Research

My research area is geometric group theory, mostly the behavior of surface groups in negatively and postively curved groups. Listed below are publications and preprints, in roughly reverse chronological order. Click or scroll down for abstracts and files.

[Mathematical notation on this page is rendered using MathJax.]
[Page format shamelessly stolen from Matt Clay.]

In Progress
Graph small cancellation groups are automatic (with Trey Trampel)

Publications
Bounding surface actions on hyperbolic spaces, Geometriae Dedicata.
A loop theorem/Dehn's lemma for some orbifolds, Algebraic & Geometric Topology, 11 (2011), 2815-2827.
Super-exponential 2-dimensional Dehn functions, to appear in Groups, Geometry, and Dynamics.
Super-exponential distortion of subgroups of CAT(-1) groups, Algebraic & Geometric Topology, 7 (2007), 301-308.
Distortion of surface groups in CAT(0) free-by-cyclic groups, Geometriae Dedicata, 120 (2006), no. 1, 119-139.

Thesis
Ends of word-hyperbolic three-manifolds, PhD thesis (2004)

• On tameness of surface actions on Gromov hyperbolic 3-manifolds preprint (pdf).  Abstract.   We show that any simplicial free action of the fundamental group of a closed hyperbolic surface on a contractible, uniformly locally finite simplicial, $$\delta$$-hyperbolic 3-manifold is geometrically and topologically tame. The proof is based on Bonahon's proof of geometric tameness for freely indecomposable hyperbolic 3-manifolds. (3/26/13)

• Bounding surface actions on hyperbolic spaces II preprint (pdf).  Abstract.   Suppose $$S$$ is a closed orientable surface of genus $$g\geq 2$$, and that $$\pi_1(S)$$ acts freely and properly discontinuously by isometries on a simply connected $$\delta$$-hyperbolic path-metric space $$X$$. Suppose further that the injectivity radius of this action is bounded below by $$\epsilon>0$$. There is a constant $$D$$, depending only on $$\delta$$, $$g$$, $$\epsilon$$, and $$f$$, with the following property: If $$\Gamma$$ is a graph that fills S and has $$f$$ complementary components, then there is a map $$\varphi:S\to X/\pi_1(S)$$, inducing the natural isomorphism on fundamental groups, such that the image of $$\Gamma$$ has total length no more than $$D$$. (3/14/13)

• Bounding surface actions on hyperbolic spaces to appear in Geometriae Dedicata.  Abstract.   Given an isometric action of the fundamental group of a closed orientable surface on a delta-hyperbolic space, we find a standard generating set whose translation distances are bounded above in terms of the hyperbolicity constant delta, the genus of the surface, and the injectivity radius of the action, which we assume to be strictly positive.

• A loop theorem/Dehn's lemma for some orbifolds Algebraic & Geometric Topology, 11 (2011), 2815-2827.  Abstract.   The equivariant loop theorem implies the existence of a loop theorem/Dehn's lemma for three-orbifolds that are good (covered by a three-manifold). In this note we prove a loop theorem/Dehn's lemma for any locally-orientable three-dimensional orbifold (good or bad) whose singular set is labeled with powers of 2. The proof is modeled on the standard tower construction.

• Super-exponential 2-dimensional Dehn functions (with Noel Brady and Pallavi Dani) Groups, Geometry, and Dynamics, 6 (2012), no. 1, 1-51.  Abstract.   We produce examples of groups of type $$\mathcal{F}_3$$ with 2-dimensional Dehn functions of the form exp$$^n(x)$$ (a tower of exponentials of height $$n$$), where $$n$$ is any natural number.