About
I am a mathematician working in the area of geometric group theory. I will soon begin working as a postdoc at ICMAT in Madrid. Previously, I completed my PhD at the University of Oxford under the supervision of Dawid Kielak.
CV (last updated August 2025).
Research interests
I am interested in infinite discrete groups and group cohomology. More specifically, I like \(L^2\)-invariants, their generalisations in positive characteristic, questions about the group ring (such as Kaplansky's zero divisor conjecture and Baer's conjecture), algebraic fibring, Novikov (co)homology, and coherence.Preprints and publications
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Novikov cohomology, finite domination, and cohomological dimension -
Virtual fibring of Poincaré-duality groups -
Appendix to
Profinite properties of Coxeter groups by Sam Hughes, Philip Möller, and Olga Varghese -
Dimension drop in residual chains -
On the cohomological dimension of kernels of maps to \(\mathbb Z\) -
The Hanna Neumann Conjecture for graphs of free groups with cyclic edge groups -
Division rings for group algebras of virtually compact special groups and \(3\)-manifold groups -
Homological growth of Artin kernels in positive characteristic -
Algebraic fibring of a hyperbolic \(7\)-manifold -
Improved algebraic fibrings
We introduce the \(\Sigma^*\)-invariant of a group of finite type, which is defined to be the subset of non-zero characters \(\chi \in \mathrm H^1(G;\mathbb R)\) with vanishing associated top-dimensional Novikov cohomology. We prove an analogue of Sikorav's Theorem for this invariant, namely that \(\mathrm{cd}(\ker \chi) = \mathrm{cd}(G) - 1\) if and only if \(\pm \chi \in \Sigma^*(G)\) for integral characters \(\chi\). This implies that cohomological dimension drop is an open property among integral characters. We also study the cohomological dimension of arbitrary co-Abelian subgroups. The techniques yield a short new proof of Ranicki's criterion for finite domination of infinite cyclic covers, and in a different direction, we prove that the algebra of affiliated operators \(\mathcal U(G)\) of a RFRS group \(G\) has weak dimension at most one if and only if \(G\) is an iterated (cyclic or finite) extension of a free group.
We show that a RFRS Poincaré-duality group \(G\) admits a virtual epimorphism to the integers whose kernel is itself a Poincaré-duality group over every field if and only if the \(L^2\)-homology of G vanishes and so do the positive-characteristic variants thereof.
Our investigations yield a more general relationship between cohomology at infinity of groups that algebraically fibre and their fibres. In particular, we show that if the fundamental group of an aspherical manifold of dimension at least three algebraically fibres, then the fibre is one ended.
Our investigations yield a more general relationship between cohomology at infinity of groups that algebraically fibre and their fibres. In particular, we show that if the fundamental group of an aspherical manifold of dimension at least three algebraically fibres, then the fibre is one ended.
We prove a number of results about profinite completions of Coxeter groups. For example we prove Coxeter groups are good in the sense of Serre and that various splittings of Coxeter groups arising from actions on trees are detected by the profinite completion. As an application we prove a number of families Coxeter groups are profinitely rigid amongst Coxeter groups. We also prove that Gromov-hyperbolic FC type, large type, and odd Coxeter groups are almost profinitely rigid amongst Coxeter groups. In the appendix, Sam Fisher and Sam Hughes show that the Atiyah Conjecture holds for all Coxeter groups, and that \(\ell^2\)-Betti numbers and their positive characteristic analogues are profinite invariants of Coxeter groups and of virtually compact special groups
We give a description of the Linnell division ring of a countable residually (poly-\(\mathbb{Z}\) virtually nilpotent) (RPVN) group in terms of a generalised Novikov ring, and show that vanishing top-degree cohomology of a finite type group \(G\) with coefficients in this Novikov ring implies the existence of a normal subgroup \(N \leqslant G\) such that \(\mathrm{cd}_{\mathbb Q}(N) < \mathrm{cd}_{\mathbb Q}(G)\) and \(G/N\) is poly-\(\mathbb Z\) virtually nilpotent.
As a consequence, we show that if \(G\) is an RPVN group of finite type, then its top-degree \(\ell^2\)-Betti number vanishes if and only if there is a poly-\(\mathbb Z\) virtually nilpotent quotient \(G/N\) such that \(\mathrm{cd}_{\mathbb Q}(N) < \mathrm{cd}_{\mathbb Q}(G)\). In particular, finitely generated RPVN groups of cohomological dimension \(2\) are virtually free-by-nilpotent if and only if their second \(\ell^2\)-Betti number vanishes, and therefore \(2\)-dimensional RPVN groups with vanishing second \(\ell^2\)-Betti number are coherent. As another application, we show that if \(G\) is a finitely generated parafree group with \(\mathrm{cd}(G) = 2\), then \(G\) satisfies the Parafree Conjecture if and only if the terms of its lower central series are eventually free. Note that the class of RPVN groups contains all finitely generated RFRS groups and all finitely generated residually torsion-free nilpotent groups.
As a consequence, we show that if \(G\) is an RPVN group of finite type, then its top-degree \(\ell^2\)-Betti number vanishes if and only if there is a poly-\(\mathbb Z\) virtually nilpotent quotient \(G/N\) such that \(\mathrm{cd}_{\mathbb Q}(N) < \mathrm{cd}_{\mathbb Q}(G)\). In particular, finitely generated RPVN groups of cohomological dimension \(2\) are virtually free-by-nilpotent if and only if their second \(\ell^2\)-Betti number vanishes, and therefore \(2\)-dimensional RPVN groups with vanishing second \(\ell^2\)-Betti number are coherent. As another application, we show that if \(G\) is a finitely generated parafree group with \(\mathrm{cd}(G) = 2\), then \(G\) satisfies the Parafree Conjecture if and only if the terms of its lower central series are eventually free. Note that the class of RPVN groups contains all finitely generated RFRS groups and all finitely generated residually torsion-free nilpotent groups.
For a group \(G\) of type \(\mathrm{FP}(R)\) for a ring \(R\) and a homomorphism \(\chi \colon G \rightarrow \mathbb Z\), we show that \(\mathrm{cd}_R(\mathrm{ker} \chi) = \mathrm{cd}_R(G) - 1\) if the top-dimensional cohomology of \(G\) with coefficients in the Novikov rings \(\widehat{R[G]}^{\pm\chi}\) vanishes. This criterion is applied to show 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 significantly shorter 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. A consequence of the result is that all virtually RFRS groups of rational cohomological dimension \(2\) with vanishing second \(\ell^2\)-Betti number are coherent. More generally, we show that if \(G\) is a RFRS group of cohomological dimension \(n\) and of type \(\mathrm{FP}_{n-1}(\mathbb Q)\), 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 (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 HNC was proved independently by Friedman and Mineyev in 2011. Recently, Antolín 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.
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 Zero Divisor 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 whenever \(G\) 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.
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.
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.
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 \(\mathrm{FP}_n(\mathbb{Q})\).
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 \(\mathrm{FP}_n(\mathbb{Q})\).
Links
- Oxford's Junior Topology and Group Theory Seminar, which I organised in 2023-2024
- My PhD thesis
- My master's thesis
- Some recordings of online talks:
- A Short Introduction to Profinite Groups, a talk I gave at the Finite and Residual Methods around GGT section (which I organised with Sam Hughes and Paweł Piwek of the Nearly Carbon Neutral Geometric Topology Conference)
- My talk Fibring of RFRS groups at the 2022 Workshop in Geometric Topology at TCU
- Art: bosny, Caleb Sandblom
Contact
- samuel [dot] p [dot] fisher [at] gmail [dot] com