I am interested in geometric group theory and, more broadly, infinite group theory from a not necessarily geometric perspective. I like homological methods in group theory, \(\ell^2\)-invariants (and their generalisations in positive characteristic), algebraic fibring, finiteness properties of groups, and group algebras.
Preprints and publications
On the cohomological dimension of kernels of maps to \(\mathbb Z\) (2024).
arXiv
We prove that if \(G\) is a finitely generated RFRS group of cohomological dimension \(2\), then \(G\) is virtually free-by-cyclic if and only if \(b_2^{(2)}(G) = 0\). This answers a question of Wise and generalises and gives a new proof of a recent theorem of Kielak and Linton, where the same result is obtained under the additional hypotheses that \(G\) is virtually compact special and hyperbolic. More generally, we show that if \(G\) is a RFRS group of cohomological dimension \(n\) and of type \(\mathrm{FP}_{n-1}\), then \(G\) admits a virtual map to \(\mathbb Z\) with kernel of rational cohomological dimension \(n-1\) if and only if \(b_n^{(2)}(G) = 0\).
The Hanna Neumann Conjecture for graphs of free groups with cyclic edge groups (2023). Joint with Ismael Morales.
Submitted. arXiv
The Hanna Neumann Conjecture (HNC) for a free group \(G\) predicts that \(\overline\chi(U\cap V)\leq \overline\chi (U)\overline\chi(V)\) for all finitely generated subgroups \(U\) and \(V\), where \(\overline\chi(H) = \min\{-\chi(H),0\}\) denotes the reduced Euler characteristic of \(H\). A strengthened version of the HNCwas proved independently by Friedman and Mineyev in 2011. Recently, Antol\'in and Jaikin-Zapirain introduced the \(L^2\)-Hall property and showed that if \(G\) is a hyperbolic limit group that satisfies this property, then \(G\) satisfies the HNC. Antolín–Jaikin-Zapirain established the \(L^2\)-Hall property for free and surface groups, which Brown–Kharlampovich extended to all limit groups. In this article, we prove the \(L^2\)-Hall property for graphs of free groups with cyclic edge groups that are hyperbolic relative to virtually abelian subgroups and also give another proof of the \(L^2\)-Hall property for limit groups. As a corollary, we show that all these groups satisfy a strengthened version of the HNC.
Division rings for group algebras of virtually compact special groups and \(3\)-manifold groups (2023). J. Comb. Algebra (to appear). Joint with Pablo Sánchez-Peralta, arXiv
Let \(k\) be a division ring and let \(G\) be either a torsion-free virtually compact special group or a torsion-free \(3\)-manifold group. We embed the group algebra \(kG\) in a division ring and prove that the embedding is Hughes-free whenever \(G\) is locally indicable. In particular, we prove that Kaplansky's zerodivisor conjecture holds for all group algebras of torsion-free \(3\)-manifold groups. The embedding is also used to confirm a conjecture of Kielak and Linton. Thanks to the work of Jaikin-Zapirain, another consequence of the embedding is that \(kG\) is coherent wehenver \(G\) is virtually compact special one-relator group.
    If \(G\) is a torsion-free one-relator group, let \(\overline{kG}\) be the division ring containing \(kG\) constructed by Lewin and Lewin. We prove that \(\overline{kG}\) is Hughes-free whenever a Hughes-free \(kG\)-division ring exists. This is always the case when \(k\) is of characteristic zero; in positive characteristic, our previous result implies this happens when \(G\) is virtually compact special.
Homological growth of Artin kernels in positive characteristic (2023). Math. Ann. (to appear). Joint with Sam Hughes and Ian J. Leary. Open access, arXiv
We prove an analogue of the Lück Approximation Theorem in positive characteristic for certain residually finite rationally soluble (RFRS) groups including right-angled Artin groups and Bestvina–Brady groups. Specifically, we prove that the mod \(p\) homology growth equals the dimension of the group homology with coefficients in a certain universal division ring and this is independent of the choice of residual chain. For general RFRS groups we obtain an inequality between the invariants. We also consider a number of applications to fibring, amenable category, and minimal volume entropy.
Algebraic fibring of a hyperbolic \(7\)-manifold (2023). Bull. Lond. Math. Soc. vol. 55, no. 3. Open access, arXiv
We show there is a finite-volume, hyperbolic \(7\)-manifold that algebraically fibres with finitely presented kernel of type \(\mathtt{FP}(\mathbb Q)\). This manifold is a finite cover of the one constructed by Italiano–Martelli–Migliorini.
We show that a virtually RFRS group \(G\) of type \(\mathrm{FP}_n(\mathbb{Q})\) virtually algebraically fibres with kernel of type \(\mathrm{FP}_n(\mathbb{Q})\) if and only if the first n \(\ell^2\)-Betti numbers of \(G\) vanish, that is, \(b^{(2)}_p(G)=0\) for \(0 \leqslant p \leqslant n\). We also offer a variant of this result over other fields, in particular in positive characteristic.
    As an application of the main result, we show that virtually amenable RFRS groups of type \(\mathrm{FP}_n(\mathbb{Q})\) are polycyclic-by-finite. It then follows that if \(G\) is a virtually RFRS group of type \(\mathrm{FP}_n(\mathbb{Q})\) such that \(\mathbb ZG\) is Noetherian, then \(G\) is polycyclic-by-finite. This answers a longstanding conjecture of Baer for virtually RFRS groups of type \(\mathrm{FP}_n(\mathbb{Q})\).
Invited talks
"Virtual fibering of manifolds and groups," Glasgow Geometry and Topology Seminar (March 2024)
"Kaplansky's zero divisor conjecture via embeddings into division rings," Southampton Pure Mathematics Colloquium (March 2024)
"Algebraic fibring and \(L^2\)-Betti numbers," McGill Geometric Group Theory Seminar (March 2023)
"Analogues of \(\ell^2\)-Betti numbers in positive characteristic and homology growth," ICMAT Group Theory Seminar (January 2023)
"\(\ell^2\)-Betti numbers and algebraic fibring," ICMAT Group Theory Seminar (January 2023)
"Agrarian invariants and a positive characteristic version of Lück's approximation conjecture," Southampton Pure Maths Lunchtime Seminar (November 2022)
"Algebraic fibring and \(\ell^2\)-Betti numbers," KIT AG Topology Seminar (November 2022)