## Robert Kropholler's Research Page

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## Published Papers

1. Hyperbolic Groups with Finitely Presented Subgroups not of Type $F_3$ (With an appendix by Giles Gardam)

Geometriae Dedicata

Published version    Arxiv version

We generalise the constructions of Brady and Lodha to give infinite families of hyperbolic groups, each having a finitely presented subgroup that is not of type $F_3$. By calculating the Euler characteristic of the hyperbolic groups constructed, we prove that infinitely many of them are pairwise non isomorphic. We further show that the first of these constructions cannot be generalised to dimensions higher than 3.

2. Right-angled Artin group boundaries (Joint with Michael Ben-Zvi)

Proceedings of the American Mathematical Society

Published version    Arxiv version

In all known examples of a CAT(0) group acting on CAT(0) spaces with non-homeomorphic CAT(0) visual boundaries, the boundaries are each not path connected. In this paper, we show this does not have to be the case by providing examples of right-angled Artin groups which exhibit non-unique CAT(0) boundaries where all of the boundaries are arbitrarily connected. We also prove a combination theorem for certain amalgams of CAT(0) groups to act on spaces with non-path connected visual boundaries. We apply this theorem to some right-angled Artin groups.

3. Uncountably many quasi-isometry classes of groups of type $FP$ (Joint with Ian Leary and Ignat Soroko)

American Journal of Mathematics

Published version    Arxiv version

Previously one of the authors constructed uncountable families of groups of type $FP$ and of $n$-dimensional Poincaré duality groups for each $n\geq4$. We strengthen these results by showing that these groups comprise uncountably many quasi-isometry classes. We deduce that for each $n\geq4$ there are uncountably many quasi-isometry classes of acyclic $n$-manifolds admitting free cocompact properly discontinuous discrete group actions.

4. Closure Properties in the Class of Multiple Context Free Groups (Joint with Davide Spriano)

Groups Complexity Cryptology

Published version    Arxiv version

We show that the class of groups with k-multiple context free word problem is closed under amalgamated free products over finite subgroups. We also show that the intersection of two context free languages need not be multiple context free.

5. Groups whose word problems are not semilinear (Joint with Robert H. Gilman and Saul Schleimer)

Groups Complexity Cryptology

Published version    Arxiv version

Suppose that G is a finitely generated group and W is the formal language of words defining the identity in G. We prove that if G is a nilpotent group, the fundamental group of a finite volume hyperbolic three-manifold, or a right-angled Artin group whose graph lies in a certain infinite class, then W is not a multiple context free language.

6. Special cube complexes (based on lectures of Piotr Przytycki)

London Mathematical Society Lecture Note Series (444): Geometric and Cohomological Group Theory

Published version

We give an account on the programme of Wise for proving residual finiteness of hyperbolic groups with quasi-convex hierarchy. The key role is played by special cube complexes introduced by Haglund and Wise. We explain the result of Haglund and Wise saying that special cube complexes are invariant under Malnormal Amalgamation. We also suggest how cubical small cancellation leads to Wise's Malnormal Special Quotient Theorem. As a consequence, all closed hyperbolic 3-manifolds with a geometrically incompressible surface are virtually special.

7. Profinite properties of RAAGs and special groups (Joint with Gareth Wilkes)

Bulletin of the London Mathematical Society

Published version    Arxiv version

In this paper we prove that RAAGs are distinguished from each other by their pro-p completions for any choice of prime p, and that RACGs are distinguished from each other by their pro-2 completions. We also give a new proof that hyperbolic virtually special groups are good in the sense of Serre. Furthermore we give an example of a property of the underlying graph of a RAAG that translates to a property of the profinite completion.

8. Non Hyperbolic Free-By-Cyclic and One-Relator Groups (Joint with Jack Button)

New York Journal of Mathematics

Published version    Arxiv version

We show that the free-by-cyclic groups of the form F(2)-by-Z act properly cocompactly on CAT(0) square complexes. We also show using generalised Baumslag-Solitar groups that all known groups defined by a 2-generator 1-relator presentation are either SQ-universal or are cyclic or isomorphic to BS(1,j). Finally we consider free-by-cyclic groups which are not relatively hyperbolic with respect to any collection of subgroups.

9. ## Accepted Papers

10. Medium-scale curvature for Cayley graphs (Joint with Assaf Bar-Natan and Moon Duchin)

Accepted to Journal of Topology and Analysis

Arxiv version

We introduce a notion of Ricci curvature for Cayley graphs that can be called medium-scale because it is neither infinitesimal nor asymptotic, but based on a chosen finite radius parameter. For this definition, abelian groups are identically flat, and on the other hand we show that $\kappa\equiv0$ implies the group is virtually abelian. In right-angled Artin groups, the curvature is zero on central elements and negative otherwise. On the other hand, we find nilpotent, CAT(0), and hyperbolic groups with points of positive curvature. We study dependence on generators and behavior under embeddings.

11. Hyperbolic groups with almost finitely presented subgroups (With an appendix joint with Federico Vigolo)

Accepted to Groups, Geometry and Dynamics

Arxiv version

In this paper we create many examples of hyperbolic groups with subgroups satisfying interesting finiteness properties. We give the first examples of subgroups of hyperbolic groups which are of type $FP_2$ but not finitely presented. We give uncountably many groups of type $FP_2$ with similar properties to those subgroups of hyperbolic groups. Along the way we create more subgroups of hyperbolic groups which are finitely presented but not of type $FP_3$.

12. Incoherence and Fibering of Many Free-by-Free groups (Joint with Genevieve Walsh)

Accepted to Annales de l'institut Fourier

Arxiv version

We show that free-by-free groups satisfying a particular homological criterion are incoherent. This class is large in nature, including many examples of hyperbolic and non-hyperbolic free-by-free groups. We apply this criterion to finite index subgroups of $F_2\rtimes F_n$ to show incoherence of all such groups, and to other similar classes of groups. Furthermore, we show that a large class of groups, including all these groups, virtually algebraically fiber.

13. Finitely generated groups acting uniformly properly on hyperbolic space (Joint with Vladimir Vankov)

Accepted to Groups, Geometry and Dynamics

Arxiv version

We construct an uncountable sequence of groups acting uniformly properly on hyperbolic spaces. We show that only countably many of these groups can be virtually torsion-free. This gives new examples of groups acting uniformly properly on hyperbolic spaces that are not virtually torsion-free and cannot be subgroups of hyperbolic groups.

14. A coarse embedding theorem for homological filling functions (Joint with Mark Pengitore)

Accepted to Bulletin of London Mathematical Society

Arxiv version

We demonstrate under appropriate finiteness conditions that a coarse embedding induces an inequality of homological Dehn functions. Applications of the main results include a characterization of what finitely presentable groups may admit a coarse embeddings into hyperbolic group of geometric dimension 2, characterizations of finitely presentable subgroups of groups with quadratic Dehn function with geometric dimension 2, and to coarse embeddings of nilpotent groups into other nilpotent groups of the same growth and into hyperbolic groups.

15. Folding-like techniques for CAT(0) cube complexes (Joint with Michael Ben-Zvi and Rylee Alanza Lyman)

Accepted to Proceedings of the Cambridge Philosophical Society

Arxiv version

In a seminal paper, Stallings introduced folding of morphisms of graphs. One consequence of folding is the representation of finitely-generated subgroups of a finite-rank free group as immersions of finite graphs. Stallings's methods allow one to construct this representation algorithmically, giving effective, algorithmic answers and proofs to classical questions about subgroups of free groups. Recently Dani--Levcovitz used Stallings-like methods to study subgroups of right-angled Coxeter groups, which act geometrically on CAT(0) cube complexes. In this paper we extend their techniques to fundamental groups of non-positively curved cube complexes.

16. ## Papers Under Review

17. Almost Hyperbolic Groups with Almost Finitely Presented Subgroups

Arxiv version

We construct new examples of CAT(0) groups containing non finitely presented subgroups that are of type $FP_2$, these CAT(0) groups do not contain copies of $\mathbb{Z}^3$. We also give a construction of groups which are of type $F_n$ but not $F_{n+1}$ with no free abelian subgroups of rank greater than $\lceil \frac{n}{3}\rceil$.

18. A new construction of CAT(0) cube complexes (Joint with Federico Vigolo)

Arxiv version

We introduce the notion of cube complex with coupled link (CLCC) as a mean of constructing interesting CAT(0) cubulated groups. CLCCs are defined locally, making them a useful tool to use when precise control over the links is required. In this paper we study some general properties of CLCCs, such as their (co)homological dimension and criteria for hyperbolicity. Some examples of fundamental groups of CLCCs are RAAGs, RACGs, surface groups and some manifold groups. As immediate applications of our criteria, we reprove the fact that RACGs are hyperbolic if and only if their defining graph is 5-large and we also provide a number of explicit examples of 3-dimensional cubulated hyperbolic groups.

19. Groups with arbitrary cubical dimension gap (Joint with Chris O'Donnell)

Arxiv version

We prove that if $G=G_1\times\dots\times G_n$ acts essentially, properly and cocompactly on a CAT(0) cube complex $X$, then the cube complex splits as a product. We use this theorem to give various examples of groups for which the minimal dimension of a cube complex the group acts on is strictly larger than that of the minimal dimension of a CAT(0) space upon which the group acts.

20. $\ell^2$ Betti numbers and coherence of random groups (Joint with Dawid Kielak and Gareth Wilkes)

Arxiv version

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.

21. Medium-scale curvature at larger radii in finitely generated groups (Joint with Brendan Mallery)

Arxiv version

In this paper we study medium scale curvature, a notion of Ricci curvature for groups. We show that dead-end elements yield non-negative curvature. We make use of related elements to find large classes of positive curvature in the lamplighter and Houghton's group. We also study the effect of increasing the radius of curvature and give examples of positive curvature for arbitrary radius. Finally we make some comparisons between this notion of curvature and the Ricci curvature introduced by Ollivier.

22. Homological Dehn functions of groups of type $FP_2$ (Joint with Noel Brady and Ignat Soroko)

Arxiv version

We prove foundational results for homological Dehn functions of groups of type $FP_2$ such as superadditivity and the invariance under quasi-isometry. We then study the homological Dehn functions of Leary's groups $G_L(S)$ providing methods to obtain uncountably many groups with a given homological Dehn function. This allows us to show that there exist groups of type $FP_2$ with quartic homological Dehn function and unsolvable word problem.

23. Extensions of hyperbolic groups have locally uniform exponential growth (Joint with Rylee Alanza Lyman and Thomas Ng)

Arxiv version

We introduce a quantitative characterization of subgroup alternatives modeled on the Tits alternative in terms of group laws and investigate when this property is preserved under extensions. We develop a framework that lets us expand the classes of groups known to have locally uniform exponential growth to include extensions of either word hyperbolic or right-angled Artin groups by groups with locally uniform exponential growth. From this, we deduce that the automorphism group of a torsion-free one-ended hyperbolic group has locally uniform exponential growth. Our methods also demonstrate that automorphism groups of torsion-free one-ended toral relatively hyperbolic groups and certain right-angled Artin groups satisfy our quantitative subgroup alternative.

24. Constructing groups of type $FP_2$ over fields but not over the integers

Arxiv version

We construct examples of groups that are $FP_2(\mathbb{Q})$ and $FP_2(\mathbb{Z}/p\mathbb{Z})$ for all primes $p$ but not of type $FP_2(\mathbb{Z})$.

25. Dehn functions of coabelian subgroups of direct products of groups (Joint with Claudio Llosa Isenrich)

Arxiv version

We develop new methods for computing the precise Dehn functions of coabelian subgroups of direct products of groups, that is, subgroups which arise as kernels of homomorphisms from the direct product onto a free abelian group. These improve and generalise previous results by Carter and Forester on Dehn functions of level sets in products of simply connected cube complexes, by Bridson on Dehn functions of cocyclic groups and by Dison on Dehn functions of coabelian groups. We then provide several applications of our methods to subgroups of direct products of free groups, to groups with interesting geometric finiteness properties and to subgroups of direct products of right-angled Artin groups.

26. Virtual algebraic fibrations of surface-by-surface groups and orbits of the mapping class group (Joint with Stefano Vidussi and Genevieve Walsh)

Arxiv version

We show that a conjecture of Putman-Wieland, which posits the nonexistence of finite orbits for higher Prym representations of the mapping class group, is equivalent to the existence of surface-by-surface and surface-by-free groups which do not virtually algebraically fiber.