I'm primarily interested in geometric group theory and topology. Recently that has meant the geometry of (groups acting on) horocyclic products.
Groups acting on horocyclic products (with Daniel Levitin) Horocyclic products are a well-studied class of metric spaces that provide models for various solvable Lie groups, Baumslag-Solitar groups, and Lamplighter groups. Let G act geometrically on a horocyclic product X⋈Y of CAT(-κ) spaces . We show that every such group is either an ascending HNN extension of a finitely-generated virtually nilpotent group, or else is not finitely presented, depending on the connectivity of the visual boundary of X⋈Y.
Solvable Baumslag-Solitar Lattices The solvable Baumslag Solitar groups BS(1,n) each admit a canonical model space, X_n. We give a complete classification of lattices in G_n = Isom^+(X_n) and find that such lattices fail to be strongly rigid---there are automorphisms of lattices H ⊂ G_n which do not extend to G_n---but do satisfy a weaker form of rigidity: for all isomorphic lattices H_1, H_2 ⊂ G_n, there is an automorphism 𝜌 ∈ Aut(G_n) so that 𝜌(H_1) = H_2.
Totally symmetric sets (with Dan Margalit) We survey the theory of totally symmetric sets, with applications to homomorphisms of symmetric groups, braid groups, linear groups, and mapping class groups.
Large totally symmetric sets (New York Journal of Mathematics, to appear) We prove that if a group has a totally symmetric set of size k, it must have order at least (k + 1)!. We also show that with three exceptions, {(1 i) | i = 2, . . . , n} ⊂ Sn is the only totally symmetric set making this bound sharp; it is thus the largest totally symmetric set relative to the size of the ambient group.
Totally symmetric sets in the general linear group (with Nick Salter, Michigan Math. J., to appear) We introduce a more general perspective on total symmetry, and formulate a notion of "irreducibility" for totally symmetric sets in the general linear group. We classify irreducible totally symmetric sets, as well as those of maximal cardinality. The supplemental Mathematica notebook can be found on Nick Salter's page.