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Non Hyperbolic Free-By-Cyclic and One-Relator Groups (Joint with Jack Button)
New York Journal of Mathematics
We show that the free-by-cyclic groups of the form F(2)-by-Z act properly cocompactly on CAT(0) square complexes. We also show using generalised Baumslag-Solitar groups that all known groups defined by a 2-generator 1-relator presentation are either SQ-universal or are cyclic or isomorphic to BS(1,j). Finally we consider free-by-cyclic groups which are not relatively hyperbolic with respect to any collection of subgroups.
Profinite properties of RAAGs and special groups (Joint with Gareth Wilkes)
Bulletin of the London Mathematical Society
In this paper we prove that RAAGs are distinguished from each other by their pro-p completions for any choice of prime p, and that RACGs are distinguished from each other by their pro-2 completions. We also give a new proof that hyperbolic virtually special groups are good in the sense of Serre. Furthermore we give an example of a property of the underlying graph of a RAAG that translates to a property of the profinite completion.
Closure Properties in the Class of Multiple Context Free Groups (Joint with Davide Spriano)
We show that the class of groups with k-multiple context free word problem is closed under amalgamated free products over finite subgroups. We also show that the intersection of two context free languages need not be multiple context free.
Medium-scale curvature for Cayley graphs (Joint with Assaf Bar-Natan and Moon Duchin)
We introduce a notion of Ricci curvature for Cayley graphs that can be called medium-scale because it is neither infinitesimal nor asymptotic, but based on a chosen finite radius parameter. For this definition, abelian groups are identically flat, and on the other hand we show that κ≡0 implies the group is virtually abelian. In right-angled Artin groups, the curvature is zero on central elements and negative otherwise. On the other hand, we find nilpotent, CAT(0), and hyperbolic groups with points of positive curvature. We study dependence on generators and behavior under embeddings.
Uncountably many quasi-isometry classes of groups of type $FP$ (Joint with Ian Leary and Ignat Soroko)
Previously one of the authors constructed uncountable families of groups of type $FP$ and of $n$-dimensional Poincaré duality groups for each $n\geq4$. We strengthen these results by showing that these groups comprise uncountably many quasi-isometry classes. We deduce that for each $n\geq4$ there are uncountably many quasi-isometry classes of acyclic $n$-manifolds admitting free cocompact properly discontinuous discrete group actions.