My research is in the areas of geometric group theory and low-dimensional topology. Broadly speaking, I am interested in groups that arise naturally in geometry and topology and in ways that geometry and topology can be used to answer algebraic questions about groups. Most of my work has involved the notion of stable commutator length, a concept that has been the subject of a significant amount of recent interest. Stable commutator length can be studied from algebraic, topological, and functional analytic perspectives, and this contributes to its richness. My work on stable commutator length spans all of these perspectives.
My research is described in more detail in my research statement.
Digital fixed points, approximate fixed points, and universal functions, with Laurence Boxer, Ozgur Ege, Ismet Karaca, and Jonathan Lopez, Appl. Gen. Topol. 17 (2016), 159-172.
Abstract: A. Rosenfeld introduced the notion of a digitally continuous function between digital images, and showed that although digital images need not have fixed point properties analogous to those of the Euclidean spaces modeled by the images, there are often approximate fixed point properties of such images. In the current paper, we obtain additional results concerning fixed points and approximate fixed points of digitally continuous functions. Among these are several results concerning the relationship between universal functions and the approximate fixed point property (AFPP).
Stable commutator length in Baumslag-Solitar groups and quasimorphisms for tree actions, with Matt Clay and Max Forester, Trans. Amer. Math. Soc. 368 (2016) no. 7, 4751-4785.
Abstract: This paper has two parts, on Baumslag-Solitar groups and on general G-trees.
In the first part we establish bounds for stable commutator length (scl) in Baumslag-Solitar groups. For a certain class of elements, we further show that scl is computable and takes rational values. We also determine exactly which of these elements admit extremal surfaces.
In the second part we establish a universal lower bound of 1/12 for scl of suitable elements of any group acting on a tree. This is achieved by constructing efficient quasimorphisms. Calculations in the group BS(2,3) show that this is the best possible universal bound, thus answering a question of Calegari and Fujiwara. We also establish scl bounds for acylindrical tree actions.
Returning to Baumslag-Solitar groups, we show that their scl spectra have a uniform gap: no element has scl in the interval (0, 1/12).
Abstract: A hyperbolic conjugacy class in the modular group PSL(2,Z) corresponds to a closed geodesic in the modular orbifold. Some of these geodesics virtually bound immersed surfaces, and some do not; the distinction is related to the polyhedral structure in the unit ball of the stable commutator length norm. We prove the following stability theorem: for every hyperbolic element of the modular group, the product of this element with a sufficiently large power of a parabolic element is represented by a geodesic that virtually bounds an immersed surface.