Here is a list of my preprints, publications, and other writing (click the titles for abstracts).
Links to my collaborators websites can be found at the bottom of the page.
(With Benjamin Brück, Dawid Kielak and Piotr Mizerka)
Non-vanishing unitary cohomology of low-rank integral special linear groups.
arXiv,
pdf,
Zenodo.
We construct explicit finite-dimensional orthogonal representations \(\pi_N\) of \(\mathrm{SL}_{N}(\mathbb Z)\) for \(N\in \{3,4\}\) all of whose invariant vectors are trivial, and such that \(H^{N - 1}(\mathrm{SL}_{N}(\mathbb Z),\pi_N)\) is non-trivial. This implies that for \(N\) as above, the group \(\mathrm{SL}_{N}(\mathbb Z)\) does not have property \((T_{N-1})\) of Bader--Sauer and therefore is not \((N-1)\)-Kazhdan in the sense of De Chiffre--Glebsky--Lubotzky--Thom, both being higher versions of Kazhdan's property \(T\).
(With Samuel M. Corson, Phillip Möller and Olga Varghese)
Profinite rigidity of affine Coxeter groups, submitted.
arXiv,
pdf.
We prove that affine Coxeter groups are profinitely rigid.
(With Daniel Ruberman)
Simple groups and complements of smooth surfaces in simply connected \(4\)-manifolds, submitted.
arXiv,
pdf,
Quanta Magazine.
For each integer \(n\) we construct a simply connected \(4\)-manifold \(X\) admitting a smoothly embedded
surface \(\Sigma\) of self intersection number \(n\) such that the complement of the surface has
non-trivial fundamental group. This answers a question of Kronheimer in Kirby's 1997 problem
list. The proof combines a topological construction with homological properties of simple
groups such as Thompson's group \(V\) and certain sporadic finite simple groups.
(With Dawid Kielak)
BNSR invariants and \(\ell^2\)-homology, submitted.
arXiv,
pdf.
We prove that if the nth \(\ell^2\)-Betti number of a group is non-zero then its nth BNSR invariant over \(\mathbb{Q}\) is empty, under suitable finiteness
conditions. We apply this to answer questions of Friedl--Vidussi and Llosa Isenrich--Py about aspherical Kähler manifolds, to verify some cases of the
Singer Conjecture, and to compute certain BNSR invariants of poly-free and poly-surface groups.
(With Naomi Andrew and Yassine Guerch)
Automorphisms of relatively hyperbolic groups and the Farrell--Jones Conjecture, submitted.
arXiv,
pdf.
We prove the fibred Farrell--Jones Conjecture (FJC) in \(A\)-, \(K\)-, and \(L\)-theory for a large class of suspensions of relatively hyperbolic groups, as well as for all suspensions of one-ended hyperbolic groups. We deduce two applications:
(1) FJC for the automorphism group of a one-ended group hyperbolic relative to virtually polycyclic subgroups;
(2) FJC is closed under extensions of FJC groups with kernel in a large class of relatively hyperbolic groups.
Along the way we prove a number of results about JSJ decompositions of relatively hyperbolic groups which may be of independent interest.
(With Samuel M. Corson, Phillip Möller and Olga Varghese)
Higman--Thompson groups and profinite properties of right-angled Coxeter groups, submitted.
arXiv,
pdf.
We prove that every right-angled Coxeter group (RACG) is profinitely rigid amongst all Coxeter groups.
On the other hand we exhibit RACGs which have infinite profinite genus amongst all finitely
generated residually finite groups. We also establish profinite rigidity results for graph
products of finite groups. Along the way we prove that the Higman--Thompson groups \(V_n\)
are generated by \(4\) involutions, generalising a classical result of Higman for Thompson's
group \(V\).
(With Yassine Guerch and Luis Jorge Sánchez Saldaña)
Centralisers and the virtually cyclic dimension of \(\mathrm{Out}(F_N)\), submitted.
arXiv,
pdf.
We prove that the virtually cyclic (geometric) dimension of the finite index congruence
subgroup \(\mathrm{IA}_N(3)\) of \(\mathrm{Out}(F_N)\) is \(2N-2\). From this we deduce
the virtually cyclic dimension of \(\mathrm{Out}(F_N)\) is finite. Along the way we
prove Lück's property (C) holds for \(\mathrm{Out}(F_N)\), we prove that the commensurator
of a cyclic subgroup of \(\mathrm{IA}_N(3)\) equals its centraliser, we give a weaker
\(\mathrm{IA}_N(3)\) analogue of various exact sequences arising from reduction systems
for mapping class groups, and give a near complete description of centralisers of
infinite order elements in \(\mathrm{IA}_3(3)\).
(With Naomi Andrew, Yassine Guerch, and Monika Kudlinska)
Homology growth of polynomially growing mapping tori, submitted.
arXiv,
pdf,
NCNGT23 postcard,
Naomi's talk at NCNGT23
(part 1,
part 2).
We prove that residually finite mapping tori of polynomially growing automorphisms of hyperbolic groups,
groups hyperbolic relative to finitely many virtually polycyclic groups, right-angled Artin groups
(when the automorphism is untwisted), and right-angled Coxeter groups have the cheap rebuilding property
of Abert, Bergeron, Fraczyk, and Gaboriau. In particular, their torsion homology growth vanishes for
every Farber sequence in every degree.
(With Motiejus Valiunas)
A note on asynchronously automatic groups and notions of non-positive curvature, submitted.
arXiv,
pdf.
We prove groups acting cocompactly on locally finite trees with hyperbolic vertex stabilisers are
asynchronously automatic. Combining this with previous work of the authors we obtain an
example of a group satisfying several non-positive curvature properties (being a
\(\mathrm{CAT}(0)\) group, an injective group, a hierarchically hyperbolic group,
and having quadratic Dehn function) which is asynchronously automatic but not biautomatic.
(With Monika Kudlinska)
On profinite rigidity amongst free-by-cyclic groups I: the generic case, submitted.
arXiv,
pdf,
talk at CRM,
talk notes (CRM),
Monika's talk at NCNGT23
(part 1,
part 2).
We prove that amongst the class of free-by-cyclic groups, Gromov hyperbolicity is an
invariant of the profinite completion. We show that whenever \(G\) is a free-by-cyclic
group with first Betti number equal to one, and \(H\) is a free-by-cyclic group which is
profinitely isomorphic to \(G\), the ranks of the fibres and the characteristic polynomials
associated to the monodromies of \(G\) and \(H\) are equal. We further show that for
hyperbolic free-by-cyclic groups with first Betti number equal to one, the stretch
factors of the associated monodromy and its inverse is an invariant of the profinite
completion. We deduce that irreducible free-by-cyclic groups with first Betti number
equal to one are almost profinitely rigid amongst irreducible free-by-cyclic groups.
We use this to prove that generic free-by-cyclic groups are almost profinitely rigid
amongst free-by-cyclic groups. We also show a similar results for
{universal Coxeter\}-by-cyclic groups.
-
Irreducible lattices fibring over the circle, submitted.
arXiv,
pdf.
We investigate the virtual fibrings of irreducible uniform lattices. In the case of a direct product of a tree and a Euclidean space we show that virtual algebraic fibring of a given uniform lattice is equivalent to reducibility. On the other hand we construct irreducible lattices which admit maps to the integers whose kernels finiteness properties are determined by the finiteness properties of certain Bestvina--Brady groups.
-
Graphs and complexes of lattices, submitted.
arXiv,
pdf,
poster.
We study lattices acting on \(\mathrm{CAT}(0)\) spaces via their commensurated subgroups. To do this we introduce the notions of a graph of lattices
and a complex of lattices giving graph and complex of group splittings of \(\mathrm{CAT}(0)\) lattices. Using this framework we characterise
irreducible uniform \((\mathrm{Isom}(\mathbb{E}^n)\times T)\)-lattices by \(C^\ast\)-simplicity and give a necessary condition for lattices in products
with a Euclidean factor to be biautomatic. We also construct non-residually finite uniform lattices acting on arbitrary products of
right-angled buildings and non-biautomatic lattices acting on the product of \(\mathbb{E}^n\) and a right-angled building.
(With Patrick S. Nairne and Davide Spriano)
Regularity of quasigeodesics characterises hyperbolicity, to appear in Proceedings of the Royal Society of Edinburgh Section A: Mathematics.
arXiv,
pdf.
We characterise hyperbolic groups in terms of quasigeodesics in the Cayley graph forming
regular languages. We also obtain a quantitative characterisation of hyperbolicity
of geodesic metric spaces by the non-existence of certain local (3,0)-quasigeodesic
loops. As an application we make progress towards a question of Shapiro regarding
groups admitting a uniquely geodesic Cayley graph.
(With Dawid Kielak)
Profinite rigidity of fibring, to appear in Revista Matemática Iberoamericana.
arXiv,
pdf,
Oberwolfach report (by Dawid).
We introduce the classes of TAP groups, in which various types of algebraic fibring are detected
by the non-vanishing of twisted Alexander polynomials. We show that finitely presented LERF groups
lie in the class \(\mathsf{TAP}_1(R)\) for every integral domain \(R\), and deduce that
algebraic fibring is a profinite property for such groups. We offer stronger results for
algebraic fibring of products of limit groups, as well as applications to profinite rigidity
of Poincaré duality groups in dimension \(3\) and RFRS groups.
(With Dawid Kielak, Ian J. Leary, and Peter H. Kropholler)
Coherence for elementary amenable groups. Proceedings of the American Mathematical Society, 152(3), 977–986, 2024.
Journal,
arXiv,
pdf,
blog post (by Alexander Engel).
We prove that for an elementary amenable group, coherence of the group, homological coherence of the group,
and coherence of the integral group ring are all equivalent. This generalises a result of Bieri and
Strebel for finitely generated soluble groups.
(With Sam P. Fisher and Ian J. Leary)
Homological growth of Artin kernels in positive characteristic. Mathematische Annalen 389, 819–843, 2024.
Journal (open access),
arXiv,
pdf,
talk at topological complexity seminar.
We prove an analogue of the Lück Approximation Theorem in positive characteristic for certain residually finite rationally
soluble 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.
(With Naomi Andrew and Monika Kudlinska)
Torsion homology growth of polynomially growing free-by-cyclic groups. Rocky Mountain Journal of Mathematics 54(4): 933–941, 2024.
Journal,
arXiv,
pdf.
We show that the homology torsion growth of a free-by-cyclic group with polynomially growing monodromy vanishes in every
dimension independently of the choice of Farber chain. It follows that the integral torsion \(\rho^{\mathbb{Z}}\) equals the
\(\ell^2\)-torsion \(\rho^{(2)}\) verifying a conjecture of Lück for these groups.
-
(With Motiejus Valiunas) Commensurating HNN-extensions:
hierarchical hyperbolicity and biautomaticity. Commentarii Mathematici Helvetici 99(2), 397–436, 2024.
Journal (open access),
arXiv,
pdf,
talk at NCNGT22
(part 1,
part 2),
Motiejus's talk at NCNGT22
(part 1,
part 2),
talk at HHG quotients workshop.
We construct a \(\mathrm{CAT}(0)\) hierarchically hyperbolic group (HHG) acting geometrically on the product of a hyperbolic plane and a
locally-finite tree which is not biautomatic. This gives the first example of an HHG which is not biautomatic, the
first example of a non-biautomatic \(\mathrm{CAT}(0)\) group of flat-rank \(2\), and the first example of an HHG which is coarsely
injective but not Helly. Our proofs heavily utilise the space of geodesic currents for a hyperbolic surface.
-
(With Eduardo Martínez-Pedroza) Hyperbolically embedded subgroups and quasi-isometries of pairs.
Canadian Mathematical Bulletin, 66(3), 827–843, 2023.
Journal (open access),
arXiv,
pdf.
We give technical conditions for a quasi-isometry of pairs to preserve a subgroup being hyperbolically
embedded. We consider applications to the quasi-isometry and commensurability invariance of
acylindrical hyperbolicity of finitely generated groups.
(With Eduardo Martínez-Pedroza and Luis Jorge Sánchez Saldaña) Quasi-isometry invariance of
relative filling functions (appendix by Ashot Minasyan). Groups, Geometry, and Dynamics 17(4), 1483–1515, 2023.
Journal (open access),
arXiv,
pdf.
For a finitely generated group \(G\) and collection of subgroups \(\mathcal{P}\) we prove that the relative Dehn function of a pair \((G,\mathcal{P})\) is invariant under quasi-isometry
of pairs. Along the way we show quasi-isometries of pairs preserve almost malnormality of the collection and fineness of the associated coned off Cayley graphs. We also prove that
for a cocompact simply connected combinatorial \(G\)-\(2\)-complex \(X\) with finite edge stabilisers, the combinatorial Dehn function is well-defined if and only if the \(1\)-skeleton
of \(X\) is fine.
We also show that if \(H\) is a hyperbolically embedded subgroup of a finitely presented group \(G\), then the relative Dehn function of the
pair (G,H) is well-defined. In the appendix, it is shown that show that the Baumslag-Solitar group \(\mathrm{BS}(k,l)\) has a well-defined Dehn
function with respect to the cyclic subgroup generated by the stable letter if and only if neither \(k\) divides \(l\) nor \(l\) divides \(k\).
-
Cohomology of Fuchsian groups and non-Euclidean crystallographic groups.
Manuscripta Mathematica 170, 659–676, 2023.
Journal (open access),
arXiv,
pdf.
For each geometrically finite 2-dimensional non-Euclidean crystallographic group (NEC group), we compute the cohomology
groups. In the case where the group is a Fuchsian group, we also determine the ring structure of the cohomology.
-
Lattices in a product of trees, hierarchically hyperbolic groups, and virtual torsion-freeness.
Bulletin of the London Mathematical Society 54(4), 1413–1419, 2022.
Journal (open access),
arXiv,
pdf.
We prove that a group acting geometrically on a product of proper minimal \(\textrm{CAT(-1)}\) spaces without
permuting isometric factors is a hierarchically hyperbolic group. As an application we construct hierarchically
hyperbolic groups which are not virtually torsion-free.
(With Kevin Li)
Higher topological complexity of hyperbolic groups. Journal of Applied and Computational Topology 6(3), 323–329, 2022.
Journal (open access),
arXiv,
pdf.
We prove for non-elementary torsion-free hyperbolic groups \(\Gamma\) and all \(r\ge 2\) that the higher topological complexity
\({\sf{TC}}_r(\Gamma)\) is equal to \(r\cdot \mathrm{cd}(\Gamma)\). In particular, hyperbolic groups satisfy the rationality conjecture
on the \(\sf{TC}\)-generating function, giving an affirmative answer to a question of Farber and Oprea. More generally, we consider
certain toral relatively hyperbolic groups.
A note on the rational homological dimension
of lattices in positive characteristic. Glasgow Mathematical Journal 65(1), 138–140, 2022.
Journal,
arXiv,
pdf.
We show via \(\ell^2\)-homology that the rational homological dimension of a lattice in a
product of simple simply connected Chevalley groups over global function fields is
equal to the rational cohomological dimension and to the dimension of the associated
Bruhat--Tits building.
-
(With Indira Chatterji
and Peter H. Kropholler) Groups
acting on trees and the first \(\ell^2\)-Betti number.
Proceedings of the Edinburgh Mathematical Society 64(4), 916–923, 2021.
Journal (open access),
arXiv,
pdf.
We generalise results of Thomas, Allcock, Thom-Petersen, and Kar-Niblo to the first \(\ell^2\)-Betti number of quotients of certain
groups acting on trees by subgroups with free actions on the edge sets of the graphs.
-
On the equivariant \(K\)- and \(KO\)-homology of some special linear groups.
Algebraic and Geometric Topology 21(7), 3483–3512, 2021.
Journal,
arXiv,
pdf,
talk at GROOT.
We compute the equivariant \(KO\)-homology of the classifying space for proper
actions of \(\textrm{SL}_3(\mathbb{Z})\) and \(\textrm{GL}_3(\mathbb{Z})\). We also
compute the Bredon homology and equivariant \(K\)-homology of the classifying
spaces for proper actions of \(\textrm{PSL}_2(\mathbb{Z}[\frac{1}{p}])\) and
\(\textrm{SL}_2(\mathbb{Z}[\frac{1}{p}])\) for each prime \(p\). Finally, we prove
the Unstable Gromov-Lawson-Rosenberg Conjecture for a large class of groups
whose maximal finite subgroups are odd order and have periodic cohomology.
-
(With Nick Gill)
The character table of a sharply \(5\)-transitive subgroup of the alternating group of degree \(12\).
International Journal of Group Theory 10(1), 11–30, 2021.
Journal (open access),
arXiv,
pdf.
In this paper we calculate the character table of a sharply \(5\)-transitive subgroup of \({\rm Alt}(12)\), and of a
sharply \(4\)-transitive subgroup of \( \rm{Alt}(11\)).
Our presentation of these calculations is new because we make no reference to the sporadic simple Mathieu groups, and
instead deduce the desired character tables using only the existence of the stated multiply transitive permutation representations.
