I am an Associate Professor at the Mathematical Institute of the University of Oxford and a Tutorial Fellow of Hertford College.
My research interests lie somewhere between Geometric Group Theory and the theory of group rings. I am a member of the Algebra and Topology research groups. My work is funded by the ERC Starting Grant "Fibring".
I am a member of the LMS and the EMS.
The Surface Group Conjectures are statements about recognising surface groups among one-relator groups, using either the structure of their finite-index subgroups, or all subgroups. We resolve these conjectures in the two generator case. More generally, we prove that every two-generator one-relator group with every infinite-index subgroup free is itself either free or a surface group.
We construct a new type of expanders, from measure-preserving affine actions with spectral gap on origami surfaces, in each genus $g\geqslant 1$. These actions are the first examples of actions with spectral gap on surfaces of genus $g>1$. We prove that the new expanders are coarsely distinct from the classical expanders obtained via the Laplacian as Cayley graphs of finite quotients of a group. In genus $g=1$, this implies that the Margulis expander, and hence the Gabber–Galil expander, is coarsely distinct from the Selberg expander. For the proof, we use the concept of piecewise asymptotic dimension and show a coarse rigidity result: A coarse embedding of either $\mathbb R^2$ or $\mathbb H^2$ into either $\mathbb R^2$ or $\mathbb H^2$ is a quasi-isometry.
We prove that twisted $\ell^2$-Betti numbers of locally indicable groups are equal to the usual $\ell^2$-Betti numbers rescaled by the dimension of the twisting representation; this answers a question of Lück for this class of groups. It also leads to two formulae: given a fibration $E$ with base space $B$ having locally indicable fundamental group, and with a simply-connected fibre $F$, the first formula bounds $\ell^2$-Betti numbers $b^{(2)}_i(E)$ of E in terms of $\ell^2$-Betti numbers of $B$ and usual Betti numbers of $F$; the second formula computes $b^{(2)}_i(E)$ exactly in terms of the same data, provided that $F$ is a high-dimensional sphere.
We also present an inequality between twisted Alexander and Thurston norms for free-by-cyclic groups and $3$-manifolds. The technical tools we use come from the theory of generalised agrarian invariants, whose study we initiate in this paper.
We initiate the study of torsion-free algebraically hyperbolic groups; these groups generalise, and are intricately related to, groups with no Baumslag-Solitar subgroups. Indeed, for groups of cohomological dimension 2 we prove that algebraic hyperbolicity is equivalent to containing no Baumslag-Solitar subgroups. This links algebraically hyperbolic groups to two famous questions of Gromov; recent work has shown these questions to have negative answers in general, but they remain open for groups of cohomological dimension 2.
We also prove that algebraically hyperbolic groups are CSA, and so have canonical abelian JSJ-decompositions. In the two-generated case we give a precise description of the form of these decompositions.
We provide a direct connection between the $\mathcal{Z}_{\max}$ (or essential) JSJ decomposition and the Friedl--Tillmann polytope of a hyperbolic two-generator one-relator group with abelianisation of rank $2$.
We deduce various structural and algorithmic properties, like the existence of a quadratic-time algorithm computing the $\mathcal{Z}_{\max}$-JSJ decomposition of such groups.
We study $\ell^2$ Betti numbers, coherence, and virtual fibring of random groups in the few-relator model. In particular, random groups with negative Euler characteristic are coherent, have $\ell^2$ homology concentrated in dimension 1, and embed in a virtually free-by-cyclic group with high probability. Similar results are shown with positive probability in the zero Euler characteristic case.
We construct examples of fibered three-manifolds with first Betti number at least $2$ and with fibered faces all of whose monodromies extend to a handlebody.
We develop the theory of agrarian invariants, which are algebraic counterparts to $L^2$-invariants. Specifically, we introduce the notions of agrarian Betti numbers, agrarian acyclicity, agrarian torsion and agrarian polytope. We use the agrarian invariants to solve the torsion-free case of a conjecture of Friedl-Tillmann: we show that the marked polytopes they constructed for two-generator one-relator groups with nice presentations are independent of the presentations used. We also show that, for such groups, the agrarian polytope encodes the splitting complexity of the group. This generalises theorems of Friedl-Tillmann and Friedl-Lück-Tillmann. Finally, we prove that for agrarian groups of deficiency $1$, the agrarian polytope admits a marking of its vertices which controls the Bieri-Neumann-Strebel invariant of the group, improving a result of the second author and partially answering a question of Friedl-Tillmann.
We show that every oriented $n$-dimensional Poincaré duality group over a $*$-ring $R$ is amenable or satisfies a linear homological isoperimetric inequality in dimension $n-1$. As an application, we prove the Tits alternative for such groups when $n=2$. We then deduce a new proof of the fact that when $n=2$ and $R = \mathbb Z$ then the group in question is a surface group.
We prove the $K$- and $L$-theoretic Farrell-Jones Conjecture with coefficients in an additive category for every normally poly-free group, in particular for even Artin groups of FC-type, and for all groups of the form $A \rtimes \mathbb{Z}$ where $A$ is a right-angled Artin group. Our proof relies on the work of Bestvina-Fujiwara-Wigglesworth on the Farrell-Jones Conjecture for free-by-cyclic groups.
We prove that $\mathrm{Aut}(F_n)$ has Kazhdan's property (T) for every $n \geqslant 6$. Together with a previous result of Kaluba, Nowak, and Ozawa, this gives the same statement for $n\geqslant 5$.
We also provide explicit lower bounds for the Kazhdan constants of $\mathrm{SAut}(F_n)$ (with $n \geqslant 6$) and of $\mathrm{SL}_n(\mathbb Z)$ (with $n \geqslant 3$) with respect to natural generating sets. In the latter case, these bounds improve upon previously known lower bounds whenever $n> 6$.
Relying on the theory of agrarian invariants introduced in previous work, we solve a conjecture of Friedl-Tillmann: we show that the marked polytopes they constructed for two-generator one-relator groups with nice presentations are independent of the presentations used. We also show that, when the groups are additionally torsion-free, the agrarian polytope encodes the splitting complexity of the group. This generalises theorems of Friedl-Tillmann and Friedl-Lück-Tillmann.
We prove that the smallest non-trivial quotient of the mapping class group of a connected orientable surface of genus at least $3$ without punctures is $\mathrm{Sp}_{2g}(2)$, thus confirming a conjecture of Zimmermann. In the process, we generalise Korkmaz's results on $\mathbb C$-linear representations of mapping class groups to projective representations over any field.
We show that a finitely generated residually finite rationally solvable (or RFRS) group $G$ is virtually fibred, in the sense that it admits a virtual surjection to $\mathbb{Z}$ with a finitely generated kernel, if and only if the first $L^2$-Betti number of $G$ vanishes. This generalises (and gives a new proof of) the analogous result of Ian Agol for fundamental groups of $3$-manifolds.
We study the Newton polytopes of determinants of square matrices defined over rings of twisted Laurent polynomials. We prove that such Newton polytopes are single polytopes (rather than formal differences of two polytopes); this result can be seen as analogous to the fact that determinants of matrices over commutative Laurent polynomial rings are themselves polynomials, rather than rational functions. We also exhibit a relationship between the Newton polytopes and invertibility of the matrices over Novikov rings, thus establishing a connection with the invariants of Bieri-Neumann-Strebel (BNS) via a theorem of Sikorav.
We offer several applications: we reprove Thurston's theorem on the existence of a polytope controlling the BNS invariants of a 3-manifold group; we extend this result to free-by-cyclic groups, and the more general descending HNN extensions of free groups. We also show that the BNS invariants of Poincaré duality groups of type F in dimension 3 and groups of deficiency one are determined by a polytope, when the groups are assumed to be agrarian, that is their integral group rings embed in skew-fields. The latter result partially confirms a conjecture of Friedl.
We also deduce the vanishing of the Newton polytopes associated to elements of the Whitehead groups of many groups satisfying the Atiyah conjecture. We use this to show that the $L^2$-torsion polytope of Friedl-Lück is invariant under homotopy. We prove the vanishing of this polytope in the presence of amenability, thus proving a conjecture of Friedl-Lück-Tillmann.
We prove a converse to Myhill's "Garden-of-Eden" theorem and obtain in this manner a characterization of amenability in terms of cellular automata: "A group G is amenable if and only if every cellular automaton with carrier G that has gardens of Eden also has mutually erasable patterns."
This answers a question by Schupp, and solves a conjecture by Ceccherini-Silberstein, Machì and Scarabotti.
An appendix by Dawid Kielak proves that group rings without zero divisors are Ore domains precisely when the group is amenable, answering a conjecture attributed to Guba.
We show that the smallest non-abelian quotient of $\mathrm{Aut}(F_n)$ is $\mathrm{PSL}_n(\mathbb Z/2 \mathbb Z)=\mathrm L_n(2)$, thus confirming a conjecture of Mecchia-Zimmermann. In the course of the proof we give an exponential (in $n$) lower bound for the cardinality of a set on which $\mathrm{SAut}(F_n)$, the unique index $2$ subgroup of $\mathrm{Aut}(F_n)$, can act non-trivially. We also offer new results on the representation theory of $\mathrm{SAut}(F_n)$ in small dimensions over small, positive characteristics, and on rigidity of maps from $\mathrm{SAut}(F_n)$ to finite groups of Lie type and algebraic groups in characteristic $2$.
We prove that if a quasi-isometry of warped cones is induced by a map between the base spaces of the cones, the actions must be conjugate by this map. The converse is false in general, conjugacy of actions is not sufficient for quasi-isometry of the respective warped cones. For a general quasi-isometry of warped cones, using the asymptotically faithful covering constructed in a previous work with Jianchao Wu, we deduce that the two groups are quasi-isometric after taking Cartesian products with suitable powers of the integers.
Secondly, we characterise geometric properties of a group (coarse embeddability into Banach spaces, asymptotic dimension, property A) by properties of the warped cone over an action of this group. These results apply to arbitrary asymptotically faithful coverings, in particular to box spaces. As an application, we calculate the asymptotic dimension of a warped cone, improve bounds by Szabó, Wu, and Zacharias and by Bartels on the amenability dimension of actions of virtually nilpotent groups, and give a partial answer to a question of Willett about dynamic asymptotic dimension.
In the appendix, we justify optimality of the aforementioned result on general quasi-isometries by showing that quasi-isometric warped cones need not come from quasi-isometric groups, contrary to the case of box spaces.
We prove Nielsen realisation for finite subgroups of the groups of untwisted outer automorphisms of RAAGs in the following sense: given any graph $\Gamma$, and any finite group $G \leqslant \mathrm{U}^0(A_\Gamma)$ $\leqslant \mathrm{Out}^0(A_\Gamma)$, we find a non-positively curved cube complex with fundamental group $A_\Gamma$ on which $G$ acts by isometries, realising the action on $A_\Gamma$.
We investigate Friedl-Lück's universal $L^2$-torsion for descending HNN extensions of finitely generated free groups, and so in particular for $F_n$-by-$\mathbb Z$ groups. This invariant induces a semi-norm on the first cohomology of the group which is an analogue of the Thurston norm for $3$-manifold groups. We prove that this Thurston semi-norm is an upper bound for the Alexander semi-norm defined by McMullen, as well as for the higher Alexander semi-norms defined by Harvey. The same inequalities are known to hold for $3$-manifold groups. We also prove that the Newton polytopes of the universal $L^2$-torsion of a descending HNN extension of $F_2$ locally determine the Bieri-Neumann-Strebel invariant of the group. We give an explicit means of computing the BNS invariant for such groups. As a corollary, we prove that the Bieri-Neumann-Strebel invariant of a descending HNN extension of $F_2$ has finitely many connected components. When the HNN extension is taken over Fn along a polynomially growing automorphism with unipotent image in $\mathrm{GL}(n,\mathbb Z)$, we show that the Newton polytope of the universal $L^2$-torsion and the BNS invariant completely determine one another. We also show that in this case the Alexander norm, its higher incarnations, and the Thurston norm all coincide.
We generalise the Karrass-Pietrowski-Solitar and the Nielsen realisation theorems from the setting of free groups to that of free products. As a result, we obtain a fixed point theorem for finite groups of outer automorphisms acting on the relative free splitting complex of Handel-Mosher and on the outer space of a free product of Guirardel-Levitt, as well as a relative version of the Nielsen realisation theorem, which in the case of free groups answers a question of Karen Vogtmann. We also prove Nielsen realisation for limit groups, and as a byproduct obtain a new proof that limit groups are CAT(0). The proofs rely on a new version of Stallings' theorem on groups with at least two ends, in which some control over the behaviour of virtual free factors is gained.
We determine the precise conditions under which $\mathrm{SOut}(F_n)$, the unique index two subgroup of $\mathrm{Out}(F_n)$, can act non-trivially via outer automorphisms on a RAAG whose defining graph has fewer than $\frac 1 2 \binom n 2$ vertices.
We also show that the outer automorphism group of a RAAG cannot act faithfully via outer automorphisms on a RAAG with a strictly smaller (in number of vertices) defining graph.
Along the way we determine the minimal dimensions of non-trivial linear representations of congruence quotients of the integral special linear groups over algebraically closed fields of characteristic zero, and provide a new lower bound on the cardinality of a set on which $\mathrm{SOut}(F_n)$ can act non-trivially.
We show that braid groups with at most $6$ strands are CAT(0) using the close connection between these groups, the associated non-crossing partition complexes and the embeddability of their diagonal links into spherical buildings of type $A$. Furthermore, we prove that the orthoscheme complex of any bounded graded modular complemented lattice is CAT(0), giving a partial answer to a conjecture of Brady and McCammond.
We show that any finitely generated group $F$ with infinitely many ends is not a group of fractions of any finitely generated proper subsemigroup $P$, that is $F$ cannot be expressed as a product $PP^{−1}$. In particular this solves a conjecture of Navas in the positive. As a corollary we obtain a new proof of the fact that finitely generated free groups do not admit isolated left-invariant orderings.
We study homomorphisms from $\mathrm{Out}(F_3)$ to $\mathrm{Out}(F_5)$, and $\mathrm{GL}(m,K)$ for $m<7$, where $K$ is a field of characteristic other than $2$ or $3$. We conclude that all $K$-linear representations of dimension at most $6$ of $\mathrm{Out}(F_3)$ factor through $\mathrm{GL}(3,\mathbb Z)$, and that all homomorphisms from $\mathrm{Out}(F_3)$ to $\mathrm{Out}(F_5)$ have finite image.
We study the existence of homomorphisms between $\mathrm{Out}(F_n)$ and $\mathrm{Out}(F_m)$ for $n > 5$ and $m < n(n-1)/2$, and conclude that if $m$ is not equal to $n$ then each such homomorphism factors through the finite group of order $2$. In particular this provides an answer to a question of Bogopol'skii and Puga. In the course of the argument linear representations of $\mathrm{Out}(F_n)$ in dimension less than $n(n+1)/2$ over fields of characteristic zero are completely classified. It is shown that each such representation has to factor through the natural projection from $\mathrm{Out}(F_n)$ to $\mathrm{GL}(n,\mathbb Z)$ coming from the action of $\mathrm{Out}(F_n)$ on the abelianisation of $F_n$. We obtain similar results about linear representation theory of $\mathrm{Out}(F_4)$ and $\mathrm{Out}(F_5)$.
Non-crossing partitions have been a staple in combinatorics for quite some time. More recently, they have surfaced (sometimes unexpectedly) in various other contexts from free probability to classifying spaces of braid groups. Also, analogues of the non-crossing partition lattice have been introduced. Here, the classical non-crossing partitions are associated to Coxeter and Artin groups of type $\mathsf{A}_n$, which explains the tight connection to the symmetric groups and braid groups. We shall outline those developments.
For various values of $n$ and $m$ we investigate homomorphisms $\mathrm{Out}(F_n) \to \mathrm{Out}(F_m)$ and $\mathrm{Out}(F_n)\to \mathrm{GL}_m(K)$, i.e. the free and linear representations of $\mathrm{Out}(F_n)$ respectively.
By means of a series of arguments revolving around the representation theory of finite symmetric subgroups of $\mathrm{Out}(F_n)$ we prove that each homomorphism $\mathrm{Out}(F_n) \to \mathrm{GL}_m(K)$ factors through the natural map $\pi_n \colon \mathrm{Out}(F_n)\to \mathrm{GL}(H_1(F_n,\mathbb Z))\cong \mathrm{GL}_n(\mathbb Z)$ whenever $n= 3,m <7$ and $\mathrm{char}(K)\not\in \{2,3\}$, and whenever $n >5,m <\binom {n+1} 2$ and $\mathrm{char}(K) \not\in \{2,3,...,n+ 1\}$.
We also construct a new infinite family of linear representations of $\mathrm{Out}(F_n)$ (where $n >2$), which do not factor through $\pi_n$. When $n$ is odd these have the smallest dimension among all known representations of $\mathrm{Out}(F_n)$ with this property.
Using the above results we establish that the image of every homomorphism $\mathrm{Out}(F_n) \to \mathrm{Out}(F_m)$ is finite whenever $n= 3$ and $n < m <6$, and of cardinality at most $2$ whenever $n >5$ and $n < m < \binom n 2$. We further show that the image is finite when $\binom n 2 \leqslant m < \binom{n+1} 2$.
We also consider the structure of normal finite index subgroups of $\mathrm{Out}(F_n)$. If $N$ is such then we prove that if the derived subgroup of the intersection of $N$ with the Torelli subgroup $\overline{\mathrm{IA}}_n < \mathrm{Out(F_n)}$ contains some term of the lower central series of $\overline{\mathrm{IA}}_n$ then the abelianisation of $N$ is finite.
GGT Advanced Class: Grigorchuk Group
We are meeting on Fridays at 11am.
We will aim at proving two theorems: that the Grigorchuk Group is of intermediate growth, and that it violates the Atiyah Conjecture.
The lecture will now take the form of a reading group. We will be going though Cromwell's book.
The reading plan is as follows:
There is a good selection of materials helpful in learning about knots on the website of Gandalf Lechner.
We will follow the book "3-Manifolds" by John Hempel.
We will follow the notes of Holger Kammeyer. The main reference is the book `$L^2$-Invariants: Theory and Applications to Geometry and $K$-Theory' by Wolfgang Lück.