We give examples of closed orientable graph 3-manifolds having a fundamental group which is not a subgroup of GL(4, F) for any field F. This answers a question in the Kirby problem list from 1977 which is credited to the ...

We show, using acylindrical hyperbolicity, that a finitely generated group splitting over $\mathbb{Z}$ cannot be simple. We also obtain SQ-universality in most cases, for instance a balanced group (one where if two powers ...

Helfgott proved that there exists a δ > 0 such that if S is a symmetric generating subset of SL(2, p)containing 1 then either S^3=SL(2, p)or |S^3| ≥|S|^1+ δ. It is known that δ ≥ 1/3024. Here we show that δ ≤ (log_2 (7) ...

We give results on when a finitely generated torsion free group does or does not embed in SL(2, C). For instance if we glue two copies of the figure 8 knot along its torus boundary then the fundamental group of the resulting ...

We show that the free-by-cyclic groups of the form $F_2$ $\rtimes$ $\Bbb Z$ act properly cocompactly on CAT(0) square complexes. We also show using generalized Baumslag–Solitar groups that all known groups defined by a ...

We explore transversals of finite index subgroups of finitely generated groups. We show that when H is a subgroup of a rank n group G and H has index at least n in G then we can construct a left transversal for H which ...

We identify when a tubular group (the fundamental group of a finite graph of groups with $\mathbb{Z}$$^{2}$ vertex and $\mathbb{Z}$ edge groups) is free by cyclic and show, using Wise's equitable sets criterion, that every ...