The BNSR-invariants of the Houghton groups, concluded

We give a complete computation of the BNSR-invariants $\Sigma^m(H_n)$ of the Houghton groups $H_n$. Partial results were previously obtained by the author, with a conjecture about the full picture, which we now confirm. The proof involves covering relevant subcomplexes of an associated $CAT(0)$ cube complex by their intersections with certain locally convex subcomplexes, and then applying a strong form of the Nerve Lemma. A consequence of the full computation is that for each $1\le m\le n-1$, $H_n$ admits a map onto $\mathbb{Z}$ whose kernel is of type $F_{m-1}$ but not $F_m$, and moreover no such kernel is ever of type $F_{n-1}$.Comment: v2: Accepted version, to appear in Proc. Edinb. Math. Soc. 10 page

A user's guide to cloning systems

In joint work of the author with Stefan Witzel, a procedure was developed for building new examples of groups in the extended family of R. Thompson's groups, using what we termed \emph{cloning systems}. These new Thompson-like groups can be thought of as limits of families of groups, though unlike other limiting processes, e.g., direct limits, these tend to be well behaved with respect to finiteness properties. In this expository note, we distill the crucial parts of that 50-page paper into a more digestible form, for those curious to understand the construction but less curious about the gritty details. We also give one new example, of a cloning system involving signed symmetric groups.Comment: 12 pages, 4 figures. Short expository note summarizing the long paper arXiv:1405.549

On the Σ\Sigma-invariants of generalized Thompson groups and Houghton groups

We compute the higher $\Sigma$-invariants $\Sigma^m(F_{n,\infty})$ of the generalized Thompson groups $F_{n,\infty}$, for all $m,n\ge 2$. This extends the $n=2$ case done by Bieri, Geoghegan and Kochloukova, and the $m=2$ case done by Kochloukova. Our approach differs from those used in the $n=2$ and $m=2$ cases; we look at the action of $F_{n,\infty}$ on a $\textrm{CAT}(0)$ cube complex, and use Morse theory to compute all the $\Sigma^m(F_{n,\infty})$. We also obtain lower bounds on $\Sigma^m(H_n)$, for the Houghton groups $H_n$, again using actions on $\textrm{CAT}(0)$ cube complexes, and discuss evidence that these bounds are sharp.Comment: 30 pages, 6 figure

Commensurability invariance for abelian splittings of right-angled Artin groups, braid groups and loop braid groups

We prove that if a right-angled Artin group $A_\Gamma$ is abstractly commensurable to a group splitting non-trivially as an amalgam or HNN-extension over $\mathbb{Z}^n$, then $A_\Gamma$ must itself split non-trivially over $\mathbb{Z}^k$ for some $k\le n$. Consequently, if two right-angled Artin groups $A_\Gamma$ and $A_\Delta$ are commensurable and $\Gamma$ has no separating $k$-cliques for any $k\le n$ then neither does $\Delta$, so "smallest size of separating clique" is a commensurability invariant. We also discuss some implications for issues of quasi-isometry. Using similar methods we also prove that for $n\ge 4$ the braid group $B_n$ is not abstractly commensurable to any group that splits non-trivially over a "free group-free" subgroup, and the same holds for $n\ge 3$ for the loop braid group $LB_n$. Our approach makes heavy use of the Bieri--Neumann--Strebel invariant.Comment: 14 page

Representatives of elliptic Weyl group elements in algebraic groups

An element w of a Weyl group W is called elliptic if it has no eigenvalue 1 in the standard reflection representation. We determine the order of any representative g in a semisimple algebraic group G of an elliptic element w in the corresponding Weyl group W. In particular if w has order d and G is simple of type different from C_n or F_4, then g has order d in G.Comment: 20 page

Geometric structures related to the braided Thompson groups

In previous work, joint with Bux, Fluch, Marschler and Witzel, we proved that the braided Thompson groups are of type $\textrm{F}_\infty$. The proof utilized certain contractible cube complexes, which in this paper we prove are CAT(0). We then use this fact to compute the geometric invariants $\Sigma^m(F_{\textrm{br}})$ of the pure braided Thompson group $F_{\textrm{br}}$. Only the first invariant $\Sigma^1(F_{\textrm{br}})$ was previously known. A consequence of our computation is that as soon as a subgroup of $F_{\textrm{br}}$ containing the commutator subgroup $[F_{\textrm{br}},F_{\textrm{br}}]$ is finitely presented, it is automatically of type $\textrm{F}_\infty$.Comment: 22 pages, 3 figures. v2: Removed Subsection 4.3, after a mistake came to light. This subsection was unrelated to the rest of the paper, so nothing else changed in any important way

Bestvina-Brady discrete Morse theory and Vietoris-Rips complexes

We inspect Vietoris-Rips complexes $VR_t(X)$ of certain metric spaces $X$ using a new generalization of Bestvina-Brady discrete Morse theory. Our main result is a pair of metric criteria on $X$, called the Morse Criterion and Link Criterion, that allow us to deduce information about the homotopy types of certain $VR_t(X)$. One application is to topological data analysis, specifically persistence of homotopy type for certain Vietoris-Rips complexes. For example we recover some results of Adamaszek-Adams and Hausmann regarding homotopy types of $VR_t(S^n)$. Another application is to geometric group theory; we prove that any group acting geometrically on a metric space satisfying a version of the Link Criterion admits a geometric action on a contractible simplicial complex, which has implications for the finiteness properties of the group. This applies for example to asymptotically $CAT(0)$ groups. We also prove that any group with a word metric satisfying the Link Criterion in an appropriate range has a contractible Vietoris-Rips complex, and use combings to exhibit a family of groups with this property.Comment: v1: Preliminary version, comments encouraged. v2: Incorporated comments. Version to be submitted. 21 pages, 1 figur

On normal subgroups of the braided Thompson groups

We inspect the normal subgroup structure of the braided Thompson groups Vbr and Fbr. We prove that every proper normal subgroup of Vbr lies in the kernel of the natural quotient Vbr \onto V, and we exhibit some families of interesting such normal subgroups. For Fbr, we prove that for any normal subgroup N of Fbr, either N is contained in the kernel of Fbr \onto F, or else N contains [Fbr,Fbr]. We also compute the Bieri-Neumann-Strebel invariant Sigma^1(Fbr), which is a useful tool for understanding normal subgroups containing the commutator subgroup.Comment: 21 pages, 6 figures. v2: accepted version, to appear in Groups, Geometry & Dynamic

Separation in the BNSR-invariants of the pure braid groups

We inspect the BNSR-invariants $\Sigma^m(P_n)$ of the pure braid groups $P_n$, using Morse theory. The BNS-invariants $\Sigma^1(P_n)$ were previously computed by Koban, McCammond and Meier. We prove that for any $3\le m\le n$, the inclusion $\Sigma^{m-2}(P_n)\subseteq \Sigma^{m-3}(P_n)$ is proper, but $\Sigma^\infty(P_n)=\Sigma^{n-2}(P_n)$. We write down explicit character classes in each relevant $\Sigma^{m-3}(P_n)\setminus \Sigma^{m-2}(P_n)$. In particular we get examples of normal subgroups $N\le P_n$ with $P_n/N\cong\mathbb{Z}$ such that $N$ is of type $F_{m-3}$ but not $F_{m-2}$, for all $3\le m\le n$.Comment: 21 pages, 3 figure

The Basilica Thompson group is not finitely presented

We show that the Basilica Thompson group introduced by Belk and Forrest is not finitely presented, and in fact is not of type FP_2. The proof involves developing techniques for proving non-simple connectedness of certain subcomplexes of CAT(0) cube complexes.Comment: 14 pages, 4 figure