About
I am a fifth year DPhil student in mathematics at the University of Oxford working under the supervision of Dawid Kielak. I completed my undergraduate and master's degrees at McGill University. My master's thesis was supervised by Dani Wise. Here is a CV (last updated August 2024).
Research interests
I am interested in infinite groups, which I mostly study using homological techniques. I like \(\ell^2\)-invariants (and their generalisations in positive characteristic), algebraic fibring, coherence, and residual properties of groups. I am also interested in group rings of infinite groups and producing embeddings of group rings into division rings.Preprints and publications
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Dimension drop in residual chains (2024). Joint with Kevin Klinge. arXiv -
On the cohomological dimension of kernels of maps to \(\mathbb Z\) (2024). arXiv -
The Hanna Neumann Conjecture for graphs of free groups with cyclic edge groups (2023). Joint with Ismael Morales. Submitted. arXiv -
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 -
Homological growth of Artin kernels in positive characteristic . Math. Ann. 389 (2024), no. 1. Joint with Sam Hughes and Ian J. Leary. Open access, arXiv -
Algebraic fibring of a hyperbolic \(7\)-manifold . Bull. Lond. Math. Soc. 55 (2023), no. 3. Open access, arXiv -
Improved algebraic fibrings (2021). Compos. Math. Journal link, arXiv
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.
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 (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, 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 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
Contact
- fisher [at] maths.ox.ac.uk
- Office N3.10
- Mathematical Institute
- University of Oxford