arXiv subjects
Publications and source records attributed to Nicholas W. M. Touikan.
Let $Γ$ be a word hyperbolic group with a cyclic JSJ decomposition that has only rigid vertex groups, which are all fundamental groups of closed surface groups. We show that any group $H$ quasi-isometric to $Γ$ is abstractly commensurable with $Γ$.
We describe a procedure called panel collapse for replacing a CAT(0) cube complex $Ψ$ by a "lower complexity" CAT(0) cube complex $Ψ_\bullet$ whenever $Ψ$ contains a codimension-$2$ hyperplane that is extremal in one of the codimension-$1$ hyperplanes containing it. Although $Ψ_\bullet$ is not in general a subcomplex of $Ψ$, it is a subspace consisting of a subcomplex together with some cubes that sit inside $Ψ$ "diagonally". The hyperplanes of $Ψ_\bullet$ extend to hyperplanes of $Ψ$. Applying this procedure, we prove: if a group $G$ acts cocompactly on a CAT(0) cube complex $Ψ$, then there is a CAT(0) cube complex $Ω$ so that $G$ acts cocompactly on $Ω$ and for each hyperplane $H$ of $Ω$, the stabiliser in $G$ of $H$ acts on $H$ essentially. Using panel collapse, we obtain a new proof of Stallings's theorem on groups with more than one end. As another illustrative example, we show that panel collapse applies to the exotic cubulations of free groups constructed by Wise. Next, we show that the CAT(0) cube complexes constructed by Cashen-Macura can be collapsed to trees while preserving all of the necessary group actions. (It also illustrates that our result applies to actions of some non-discrete groups.) We also discuss possible applications to quasi-isometric rigidity for certain classes of graphs of free groups with cyclic edge groups. Panel collapse is also used in forthcoming work of the first-named author and Wilton to study fixed-point sets of finite subgroups of $\mathrm{Out}(F_n)$ on the free splitting complex. Finally, we apply panel collapse to a conjecture of Kropholler, obtaining a short proof under a natural extra hypothesis.
We present an algorithm which given a presentation of a group $G$ without 2-torsion, a solution to the word problem with respect to this presentation, and an acylindricity constant $κ$, outputs a collection of tracks in an appropriate presentation complex. We give two applications: the first is an algorithm which decides if $G$ admits an essential free decomposition, the second is an algorithm which; if $G$ is relatively hyperbolic; decides if it admits an essential elementary splitting.
There are limit groups having non-conjugate elements whose images are conjugate in every free quotient. Towers over free groups are freely conjugacy separable.
We introduce the notion of multipass automata as a generalization of pushdown automata and study the classes of languages accepted by such machines. The class of languages accepted by deterministic multipass automata is exactly the Boolean closure of the class of deterministic context-free languages while the class of languages accepted by nondeterministic multipass automata is exactly the class of poly-context-free languages, that is, languages which are the intersection of finitely many context-free languages. We illustrate the use of these automata by studying groups whose word problems are in the above classes.
We generalize Bulitko's Lemma to equations over (or homomorphisms into) groups that have $κ$-acylindrical splittings.
We describe the fully residually F; or limit groups relative to F; (where F is a free group) that arise from systems of equations in two variables over F that have coefficients in F.