arXiv subjects
Publications and source records attributed to Kasia Jankiewicz.
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 give criteria for a graph of groups to have finite stature with respect to its collection of vertex groups, in the sense of Huang-Wise. We apply it to the triangle Artin groups that were previously shown to split as a graph of groups. This allows us to deduce residual finiteness, and expands the list of Artin groups known to be residually finite.
We show that if $X$ is a compact nonpositively curved cube complex which is a non-product, then $π_1 X$ has a proper quotient that is $π_1$ of a cube complex with these same properties.
We show that the fundamental group of a geometrically clean graph of finite rank free groups does not need to be virtually compact special, answering a question of Wise. This implies that the class of the virtually VH-clean graphs of finite rank free groups is a proper subclass of the class of virtually geometrically clean graphs of finite rank free groups.
We prove that the Center Conjecture passes to the Artin groups whose defining graphs are cones, if the conjecture holds for the Artin group defined on the set of the cone points. In particular, it holds for every Artin group whose defining graph has exactly one cone point.
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.
Given a non-positively curved cube complex $X$, we prove that the quotient of $π_1X$ defined by a cubical presentation $\langle X\mid Y_1,\dots, Y_s\rangle$ satisfying sufficient non-metric cubical small-cancellation conditions is hyperbolic provided that $π_1X$ is hyperbolic. This generalises the fact that finitely presented classical $C(7)$ small-cancellation groups are hyperbolic.
A countable group is residually finite if every nontrivial element can act nontrivially on a finite set. When a group fails to be residually finite, we might want to measure how drastically it fails - it could be that only finitely many conjugacy classes of elements fail to act nontrivially on a finite set, or it could be that the group has no nontrivial actions on finite sets whatsoever. We define a hierarchy of properties, and construct groups which become arbitrarily complicated in this sense.
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.
We show that every group acting freely and vertex-transitively by isometries on a product of two regular trees of finite valence is boundary rigid. That means that every CAT(0) space that admits a geometric action of any such group has the visual boundary homeomorphic to a join of two copies of the Cantor set.
We show that many $2$-dimensional Artin groups are residually finite. This includes $3$-generator Artin groups with labels $\geq 4$ except for $(2m+1, 4,4)$ for any $m\geq 2$. As a first step towards residual finiteness we show that these Artin groups, and many more, split as free products with amalgamation or HNN extensions of finite rank free groups. Among others, this holds for all large type Artin groups with defining graph admitting an orientation, where each simple cycle is directed.
We give a simplified approach to the cubulation of small-cancellation quotients of free products of cubulated groups. We construct fundamental groups of compact nonpositively curved cube complexes that do not virtually split.
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.
We study uniform exponential growth of groups acting on CAT(0) cube complexes. We show that groups acting without global fixed points on CAT(0) square complexes either have uniform exponential growth or stabilize a Euclidean subcomplex. This generalizes the work of Kar and Sageev that considers free actions. Our result lets us show uniform exponential growth for certain groups that act improperly on CAT(0) square complexes, namely, finitely generated subgroups of the Higman group and triangle-free Artin groups. We also obtain that non-virtually abelian groups acting freely on CAT(0) cube complexes of any dimension with isolated flats that admit a geometric group action have uniform exponential growth.
We show that a triangle Artin group $\text{Art}_{MNP}$ where $M\leq N\leq P$ splits as an amalgamated product or an HNN extension of finite rank free groups, provided that either $M>2$, or $N>3$. We also prove that all even three generator Artin groups are residually finite.
We construct a finitely generated 2-dimensional group that acts properly on a locally finite CAT(0) cube complex but does not act properly on a finite dimensional CAT(0) cube complex.
For each $n$ we construct examples of finitely presented $C'(1/6)$ small cancellation groups that do not act properly on any $n$-dimensional CAT(0) cube complex.