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• Hyperbolic space and its isometries
• Kleinian groups (discrete groups of isometries of hyperbolic 3-space)
• Signatures of actions of automorphism groups on Riemann surfaces

The primary focus of Jim's research concerns hyperbolic space and its isometries. Specifically, Jim has spent much of his career investigating Kleinian groups, which are discrete groups of isometries of hyperbolic 3-space.

One specific question Jim has engaged with somewhat extensively is the limit set intersection theorem for Kleinian groups, which are the theorems that describe the difference between Λ(θ ∩ Φ) and Λ(θ) ∩ Λ(Φ) for subgroups θ and Φ of a Kleinian group Γ, subject to different hypotheses on Γ, θ and Φ. In dimension 3, this question is, Jim feels, essentially answered. There is still room for exploration for this question for Kleinian groups acting on higher dimensional hyperbolic space.

Another area of more recent focus is joint work with  (University of Portland), investigating signatures of actions of automorphism groups on Riemann surfaces (see below). 

At present, Jim has several projects underway:

1. Schottky groups are a particular class of Kleinian groups acting on hyperbolic 3-space, which are algebraically straightforward (as they are free groups) and dynamically interesting. Schottky groups came in two flavors, classical (which can be defined in terms of pairings of circles) and non-classical (which cannot). 

In this project, Jim investigates the extent to which subgroups of classical Schottky groups are or are not themselves classical, and the extent to which subgroups of non-classical Schottky groups are and are not themselves non-classical.

2. In joint work with (University of Portland), we continue our investigation of aspects of signatures of actions of automorphism groups acting on closed Riemann surfaces, using the framework of skeletal signatures we introduced in our earlier papers. 

3. A question that has captured Jim's imagination is the extent to which classical geometric theorems from Euclidean planar geometry hold in the hyperbolic plane.