Research interests:
My general research revolves around low-dimensional groups. I am particularly interested in one-relator groups, free-by-cyclic groups and graphs of free groups. Within these classes I mainly think about hyperbolicity, coherence, the conjugacy problem and the efficiency of the word problem. I am also interested in compression and its applications to computational questions in groups.
My articles:
Group rings of three-manifold groups (with Dawid Kielak, arXiv, to appear in Proc. Amer. Math. Soc.)
Let $G$ be the fundamental group of a three-manifold. By piecing together many known facts about three manifold groups, we establish two properties of the group ring $\mathbb{C}G$. We show that if $G$ has rational cohomological dimension two, then $\mathbb{C}G$ is coherent. We also show that if $G$ is torsion-free, then $G$ satisfies the Strong Atiyah Conjecture over $\mathbb{C}$ and hence that $\mathbb{C}G$ satisfies Kaplansky’s Zero Divisor Conjecture.
On the coherence of one-relator groups and their group algebras (with Andrei Jaikin-Zapirain, arXiv, Alexander Engel’s blog)
We prove that one-relator groups are coherent, solving a well-known problem of Gilbert Baumslag. Our proof strategy is readily applicable to many classes of groups of cohomological dimension two. Indeed we also show that fundamental groups of two-complexes with non-positive immersions are homologically coherent, that groups with staggered presentations and many Coxeter groups are coherent and we show that group algebras over fields of characteristic zero of groups with reducible presentations without proper powers are coherent.
Virtually free-by-cyclic groups (with Dawid Kielak, arXiv, Alexander Engel’s blog).
We obtain a homological characterisation of virtually free-by-cyclic groups among groups that are hyperbolic and virtually compact special. As a consequence, we show that many groups known to be coherent actually possess the stronger property of being virtually free-by-cyclic. In particular, we show that all one-relator groups with torsion are virtually free-by-cyclic, solving a conjecture of Baumslag.
Hyperbolic one-relator groups (arXiv).
We introduce two families of two-generator one-relator groups called primitive extension groups and show that a one-relator group is hyperbolic if its primitive extension subgroups are hyperbolic. This reduces the problem of characterising hyperbolic one-relator groups to characterising hyperbolic primitive extension groups. These new groups moreover admit explicit decompositions as graphs of free groups with adjoined roots. In order to obtain this result, we characterise $2$-free one-relator groups with exceptional intersection in terms of Christoffel words, show that hyperbolic one-relator groups have quasi-convex Magnus subgroups and build upon the one-relator tower machinery developed in the authors previous article.
One-relator hierarchies (arXiv, Alexander Engel’s blog).
We prove that one-relator groups with negative immersions are hyperbolic and virtually special; this resolves a recent conjecture of Louder and Wilton. The main new tool we develop is a refinement of the classic Magnus–Moldavanskii hierarchy for one-relator groups. We introduce the notions of stable HNN-extensions and stable hierarchies. We then show that a one-relator group is hyperbolic and has a quasi-convex one-relator hierarchy if and only if it does not contain a Baumslag–Solitar subgroup and has a stable one-relator hierarchy. Finally, we give an algorithm that, given as input a one-relator group, verifies if it has a stable hierarchy and, if so, determines whether or not it contains a Baumslag–Solitar subgroup.
The fully compressed subgroup membership problem (arXiv, Journal of Algebra).
Suppose that $F$ is a free group and $k$ is a natural number. We show that the fully compressed membership problem for $k$-generated subgroups of $F$ is solvable in polynomial time. In order to do this, we adapt the theory of Stallings’ foldings to handle edges with compressed labels. This partially answers a question of Markus Lohrey.
On the intersections of finitely generated subgroups of free groups: reduced rank to full rank (arXiv).
We show that the number of conjugacy classes of intersections $A\cap B^g$, for fixed finitely generated subgroups $A, B \leq F$ of a free group, is bounded above in terms of the ranks of $A$ and $B$; this confirms an intuition of Walter Neumann. This result was previously known only in the case where $A$ or $B$ is cyclic by the $w$-cycles theorem of Helfer and Wise and, independently, Louder and Wilton. In order to prove our main theorem, we introduce new results regarding the structure of fibre products of finite graphs. We work with a generalised definition of graph immersions so that our results apply to the theory of regular languages. For example, we give a new algorithm to decide non-emptiness of the intersection of two regular languages.
A database:
I am putting together a database of one-relator groups. More information can be found here.