I'm primarily interested in geometric group theory and topology. Still figuring out what I mean by that!
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.