Verma, Torsion-Free Uniform Lattices in Baumslag-Solitar Complexes (Submitted)
Abstract- Let X_{m,n} denote the combinatorial model for the Baumslag-Solitar group BS(m,n). We classify the pairs of nonzero integers (m,n) for which the locally compact group of combinatorial automorphisms, Aut(X_{m,n}), contains incommensurable torsion-free uniform lattices. In particular, we show that Aut(X_{m,n}) contains abstractly incommensurable torsion-free uniform lattices if and only if there exists a prime p ≤ gcd(m,n) such that either m/gcd(m,n) or n/gcd(m,n) is divisible by p. Additionally, we show that when incommensurable lattices do not exist in Aut(X_{m,n}), the cell complex X_{m,n} satisfies Leighton’s property.
Verma, Goswami, Superexponential distortion of a free groups in a virtually special groups (Submitted)
Abstract- For all integers $k, m > 0$, we construct a virtually special group $G$ containing a finite rank free subgroup $F$ whose distortion function in $G$ grows like $\exp^k(x^m)$. We also construct examples of virtually special groups containing finite rank free subgroups whose distortion functions grow bigger than any iterated exponential.
Verma, Lattices in Generalized Baumslag-Solitar Complexes
Abstract- In this paper, we prove that every uniform lattice in the combinatorial automorphism group of a generalized Baumslag–Solitar complex is the fundamental group of a graph of $2$-ended groups. Conversely, we show that the quotient of a graph of $2$-ended groups by a finite normal subgroup can be realized as a uniform lattice in the automorphism group of a generalized Baumslag–Solitar complex. As an application, we obtain example of a uniform lattice in the combinatorial automorphism group of a Baumslag–Solitar complex that are not virtually torsion-free.