arXiv subjects
Publications and source records attributed to Kevin Schreve.
We show that for $d\geq 2$ every finite $d$-dimensional simplicial complex is a deformation retract of a $(2d-1)$-dimensional pseudomanifold with boundary. Moreover, it embeds as a retract in a closed $(2d-1)$-dimensional pseudomanifold.
We introduce a graph-theoretic condition, called $(n,m)$--branching, that ensures a combinatorial round tree with controlled branching parameters can be quasi-isometrically embedded in the Davis complex of the right-angled Coxeter group defined by the graph. This construction yields a lower bound on the conformal dimension of the boundary of such a hyperbolic group. We exhibit numerous families of graphs with this property, including many 1-dimensional spherical buildings. We prove an embedding result, showing that under mild hypotheses a flag-no-square graph embeds as an induced subgraph in a flag-no-square triangulation of a closed surface. We use this to embed our branching graphs into graphs presenting hyperbolic right-angled Coxeter groups with Pontryagin sphere boundary. We conclude there are examples of such groups with conformal dimension tending to infinity, and hence, there are infinitely many quasi-isometry classes within this family. We use conformal dimension to show that recent work of Lafont--Minemyer--Sorcar--Stover--Wells can be upgraded to conclude that for every $n \geq 2$ there exist infinitely many quasi-isometry classes of hyperbolic right-angled Coxeter groups that virtually algebraically fiber and have virtual cohomological dimension $n$.
We show that for $n \geq 3$, there are torsion-free CAT(0) groups with visual boundaries that embed into $S^n$ but which are not virtually the fundamental group of a compact aspherical $n+1$-manifold. The groups are the CAT(0) and not bi-automatic groups constructed previously by Leary and Minasyan. The obstruction comes from analyzing certain cyclic orders on the boundary of the Bass-Serre tree, in the same manner as Kapovich-Kleiner ruled out actions of Baumslag-Solitar groups on coarse $\mathrm{PD}(3)$ spaces.
We show that Gromov-Thurston branched covers satisfy the Singer conjecture whenever the degree of the cover is not divisible by a finite set of primes determined by the base manifold and the branch locus.
If $L$ is a flag triangulation of $S^{n-1}$, then the Davis complex $Σ_L$ for the associated right-angled Coxeter group $W_L$ is a contractible $n$-manifold. A special case of a conjecture of Singer predicts that the $L^2$-homology of such $Σ_L$ vanishes outside the middle dimension. We give conditions which guarantee this vanishing is preserved under edge subdivision of $L$. In particular, we verify Singer's conjecture when $L$ is the barycentric subdivision of the boundary of an $n$-simplex, and for general barycentric subdivisions of triangulations of $S^{2n-1}$. Using this, we construct explicit counterexamples to a torsion growth analogue of Singer's conjecture.
In 2018, Kielak gave a virtual fibering criterion for RFRS groups. In this paper, we present a simpler proof of this.
Given a graph $Γ$ and a number $n$, the associated $n^{th}$ graph braid group $B_n(Γ)$ is the fundamental group of the unordered configuration space of $n$ points on $Γ$. Świątkowski showed that for a given $Γ$ and $n$ large enough, there is a free abelian subgroup of $B_n(Γ)$ of rank equal to the cohomological dimension of $B_n(Γ)$. In this note, we observe that at the cost of possibly adding additional points, we can find a subgroup of the same cohomological dimension which is a direct product of non-abelian free groups, and give some applications.
Suppose a residually finite group $G$ acts cocompactly on a contractible complex with strict fundamental domain $Q$, where the stabilizers are either trivial or have normal $\mathbb{Z}$-subgroups. Let $\partial Q$ be the subcomplex of $Q$ with nontrivial stabilizers. Our main result is a computation of the homology torsion growth of a chain of finite index normal subgroups of $G$. We show that independent of the chain, the normalized torsion limits to the torsion of $\partial Q$, shifted a degree. Under milder assumptions of acyclicity of nontrivial stabilizers, we show similar formulas for the mod p-homology growth. We also obtain formulas for the universal and the usual $L^2$-torsion of $G$ in terms of the torsion of stabilizers and topology of $\partial Q$. In particular, we get complete answers for right-angled Artin groups, which shows they satisfy a torsion analogue of the Lück approximation theorem.
We introduce a hyperbolic reflection group trick which builds closed aspherical manifolds out of compact ones and preserves hyperbolicity, residual finiteness, and -- for almost all primes $p$ -- $\mathbb{F}_p$-homology growth above the middle dimension. We use this trick, embedding theory and manifold topology to construct Gromov hyperbolic $7$-manifolds that do not virtually fiber over a circle out of graph products of large finite groups.
We compute the mod $p$ homology growth of residual sequences of finite index normal subgroups of right-angled Artin groups. We find examples where this differs from the rational homology growth, which implies the homology of subgroups in the sequence has lots of torsion. More precisely, the homology torsion grows exponentially in the index of the subgroup. For odd primes $p$, we construct closed locally CAT(0) manifolds with nonzero mod $p$ homology growth outside the middle dimension. These examples show that Singer's conjecture on rational homology growth and Lück's conjecture on torsion homology growth are incompatible with each other, so at least one of them must be wrong.
We prove that for every prime $p$ algebraically clean graphs of groups are virtually residually $p$-finite and cohomologically $p$-complete. We also prove that they are cohomologically good. We apply this to certain $2$-dimensional Artin groups.
Bregman and Clay recently characterized which right-angled Artin groups with geometric dimension 2 have vanishing minimal volume entropy. In this note, we extend this characterization to higher dimensions.
The Tits Conjecture, proved by Crisp and Paris, states that squares of the standard generators of any Artin group generate an obvious right-angled Artin subgroup. We consider a larger set of elements consisting of all the centers of the irreducible spherical special subgroups of the Artin group, and conjecture that sufficiently large powers of those elements generate an obvious right-angled Artin subgroup. This alleged right-angled Artin subgroup is in some sense as large as possible; its nerve is homeomorphic to the nerve of the ambient Artin group. We verify this conjecture for the class of locally reducible Artin groups, which includes all $2$-dimensional Artin groups, and for spherical Artin groups of any type other than $E_6$, $E_7$, $E_8$. We use our results to conclude that certain Artin groups contain hyperbolic surface subgroups, answering questions of Gordon, Long and Reid.
In this note, we prove that the $K(π,1)$-conjecture for Artin groups implies the center conjecture for Artin groups. Specifically, every Artin group without a spherical factor that satisfies the $K(π,1)$-conjecture has a trivial center.
Bestvina introduced a $\mathcal{Z}$-structure for a group $G$ to generalize the boundary of a CAT(0) or hyperbolic group. A refinement of this notion, introduced by Farrell and Lafont, includes a $G$-equivariance requirement, and is known as an $\mathcal{E}\mathcal{Z}$-structure. In this paper, we show that fundamental groups of graphs of nonpositively curved Riemannian $n$-manifolds admit $\mathcal{Z}$-structures and graphs of negatively curved or flat $n$-manifolds admit $\mathcal{E}\mathcal{Z}$-structures. This generalizes a recent result of the first two authors with Tirel, which put $\mathcal{E}\mathcal{Z}$-structures on Baumslag-Solitar groups and $\mathcal{Z}$-structures on generalized Baumslag-Solitar groups.
Whenever a finitely generated group $G$ acts properly discontinuously by isometries on a metric space $X$, there is an induced uniform embedding (a Lipschitz and uniformly proper map) $ρ: G \rightarrow X$ given by mapping $G$ to an orbit. We study when there is a difference between a finitely generated group $G$ acting properly on a contractible $n$-manifold and uniformly embedding into a contractible $n$-manifold. For example, Kapovich and Kleiner showed that there are torsion-free hyperbolic groups that uniformly embed into a contractible $3$-manifold but only virtually act on a contractible $3$-manifold. We show that $k$-fold products of these examples do not act on a contractible $3k$-manifold.
Croke and Kleiner constructed two homeomorphic locally CAT(0) complexes whose universal covers have visual boundaries that are not homeomorphic. We construct two homeomorphic locally CAT(0) complexes so that the visual boundary of one universal cover contains a nonplanar graph, while the visual boundary of the other does not. In contrast, we prove for any two locally CAT(0) metrics on the Croke-Kleiner complex, if a finite graph embeds in the visual boundary of one universal cover, then the graph embeds in the visual boundary of the other.
The action dimension of a discrete group $G$ is the minimum dimension of contractible manifold that admits a proper $G$-action. We compute the action dimension of the direct limit of a simple complex of groups for several classes of examples including: 1) Artin groups, 2) graph products of groups, and 3) fundamental groups of aspherical complements of arrangements of affine hyperplanes.