arXiv subjects
Publications and source records attributed to Daniel Groves.
We prove that if a proper metric space is quasi-isometric to a finitely generated group and to a space with a horoball over a finitely generated group, then that space is quasi-isometric to a rank-one symmetric space or the real line.
Given a hyperbolic group $G$ and a maximal infinite cyclic subgroup $\langle g \rangle$, we define a {\it drilling of $G$ along $g$}, which is a relatively hyperbolic group pair $(\widehat{G}, P)$. This is inspired by the well-studied procedure of drilling a hyperbolic $3$--manifold along an embedded geodesic. We prove that, under suitable conditions, a hyperbolic group with $2$-sphere boundary admits a drilling where the resulting relatively hyperbolic group pair $(\widehat{G}, P)$ has relatively hyperbolic boundary $S^2$. This allows us to reduce the Cannon Conjecture (in the residually finite case) to a relative version, which is likely to be more tractable.
We prove that many relatively hyperbolic groups obtained by relative strict hyperbolization admit a cocompact action on a CAT(0) cubical complex. Under suitable assumptions on the peripheral subgroups, these groups are residually finite and even virtually special. We include some applications to the theory of manifolds, such as the construction of new non-positively curved Riemannian manifolds with residually finite fundamental group, and the existence of non-triangulable aspherical manifolds with virtually special fundamental group.
We prove that for $n\geq 2$, a non-uniform lattice in $\text{PU}(n,1)$ does not admit a relatively geometric action on a $\mathrm{CAT}(0)$ cube complex, in the sense of Einstein and Groves. As a consequence, if $\Gamma$ is a non-uniform lattice in a non-compact semisimple Lie group $G$ without compact factors that admits a relatively geometric action on a $\mathrm{CAT}(0)$ cube complex, then $G$ is commensurable with $\text{SO}(n,1)$. We also prove that if a K\"ahler group is hyperbolic relative to residually finite parabolic subgroups, and acts relatively geometrically on a $\mathrm{CAT}(0)$ cube complex, then it is virtually a surface group.
Italiano-Martelli-Migliorini recently constructed hyperbolic groups which have non-hyperbolic subgroups of finite type. Using a closely related construction, Llosa Isenrich-Martelli-Py constructed hyperbolic groups with subgroups of type $F_3$ but not $F_4$. We observe that these hyperbolic groups can be chosen to be special in the sense of Haglund-Wise.
Bowditch characterized relative hyperbolicity in terms of group actions on fine hyperbolic graphs with finitely many edge orbits and finite edge stabilizers. In this paper, we define generalized fine actions on hyperbolic graphs, in which the peripheral subgroups are allowed to stabilize finite sub-graphs rather than stabilizing a point. Generalized fine actions are useful for studying groups that act relatively geometrically on a CAT(0) cube complex, which were recently defined by the first two authors. Specifically, we show that a group acting relatively geometrically on a CAT(0) cube complex admits a generalized fine action on the one-skeleton of the cube complex. For generalized fine actions, we prove a criterion for relative quasiconvexity as subgroups that cocompactly stabilize quasi-convex sub-graphs, generalizing a result of Martinez-Pedroza and Wise in the setting of fine hyperbolic graphs. As an application, we obtain a characterization of boundary separation in generalized fine graphs and use it to prove that Bowditch boundary points in relatively geometric actions are always separated by a hyperplane stabilizer.
We prove foundational results about the set of homomorphisms from a finitely generated group to the collection of all fundamental groups of compact 3-manifolds and answer questions of Reid-Wang-Zhou and Agol-Liu.
We develop the foundations of the theory of relatively geometric actions of relatively hyperbolic groups on CAT(0) cube complexes, a notion introduced in our previous work [5]. In the relatively geometric setting we prove: full relatively quasi-convex subgroups are convex compact; an analog of Agol's Theorem; and a version of Haglund--Wise's Canonical Completion and Retraction.
We prove that given any finite abelian group $A$ and any irreducible $3$-manifold $M$ with empty or toroidal boundary which is not a graph manifold there exists a finite cover $M' \to M$ so that $A$ is a direct factor in $H_1(M',\mathbb{Z})$. This generalizes results of Sun and of Friedl-Herrmann.
In arXiv:1204.2810 Agol proved the Virtual Haken and Virtual Fibering Conjectures by confirming a conjecture of Wise: Every cubulated hyperbolic group is virtually special. We extend this result to cocompactly cubulated relatively hyperbolic groups with minimal assumptions on the parabolic subgroups. Our proof proceeds by first recubulating to obtain an improper action with controlled stabilizers (a weakly relatively geometric action), and then Dehn filling to obtain many cubulated hyperbolic quotients. We apply our results to prove the Relative Cannon Conjecture for certain cubulated or partially cubulated relatively hyperbolic groups. One of our main results (Theorem A) recovers via different methods a theorem of Oreg\'on-Reyes (arXiv:2003.12702).
We introduce a new kind of action of a relatively hyperbolic group on a CAT(0) cube complex, called a relatively geometric action. We provide an application to characterize finite-volume Kleinian groups in terms of action on cube complexes, analogous to the results of Markovic and Ha\"issinsky in the closed case.
In this paper we study hyperbolic groups acting on CAT(0) cube complexes. The first main result (Theorem A) is a structural result about the Sageev construction, in which we relate quasi-convexity of hyperplane stabilizers with quasi-convexity of cell stabilizers. The second main result (Theorem D) generalizes both Agol's theorem on cubulated hyperbolic groups and Wise's Quasi-convex Hierarchy Theorem.
We define a new condition on relatively hyperbolic Dehn filling which allows us to control the behavior of a relatively quasiconvex subgroups which need not be full. As an application, in combination with a recent result of Cooper and Futer, we provide a new proof of the virtual fibering of non-compact finite-volume hyperbolic 3-manifolds, a result first proved by Wise. Additionally, we explain how the results of [2, Appendix A] can be generalized to the relative setting to control the relative height of relatively quasiconvex subgroups under appropriate Dehn fillings.
We study the set of homomorphisms from a fixed finitely generated group into a family of groups which are `uniformly acylindrically hyperbolic'. Our main results reduce this study to sets of homomorphisms which do not diverge in an appropriate sense. As an application, we prove that any relatively hyperbolic group with equationally noetherian peripheral subgroups is itself equationally noetherian.
A finitely generated subgroup H of a torsion-free hyperbolic group G is called immutable if there are only finitely many conjugacy classes of injections of H into G. We show that there is no uniform algorithm to recognize immutability, answering a uniform version of a question asked by the authors.
We begin an investigation into the behavior of Bowditch and Gromov boundaries under the operation of Dehn filling. In particular we show many Dehn fillings of a toral relatively hyperbolic group with 2-sphere boundary are hyperbolic with 2-sphere boundary. As an application, we show that the Cannon conjecture implies a relatively hyperbolic version of the Cannon conjecture.
Let $\Gamma$ be a torsion-free hyperbolic group. We study $\Gamma$--limit groups which, unlike the fundamental case in which $\Gamma$ is free, may not be finitely presentable or geometrically tractable. We define model $\Gamma$--limit groups, which always have good geometric properties (in particular, they are always relatively hyperbolic). Given a strict resolution of an arbitrary $\Gamma$--limit group $L$, we canonically construct a strict resolution of a model $\Gamma$--limit group, which encodes all homomorphisms $L\to \Gamma$ that factor through the given resolution. We propose this as the correct framework in which to study $\Gamma$--limit groups algorithmically. We enumerate all $\Gamma$--limit groups in this framework.
We consider conditions on relatively hyperbolic groups about the non-existence of certain kinds of splittings, and show these properties persist in long Dehn fillings. We deduce that certain connectivity properties of the Bowditch boundary persist under long fillings.