Combing Euclidean buildings

For an arbitrary Euclidean building we define a certain combing, which satisfies the `fellow traveller property' and admits a recursive definition. Using this combing we prove that any group acting freely, cocompactly and by order preserving automorphisms on a Euclidean building of one of the types A_n,B_n,C_n admits a biautomatic structure.Comment: 32 pages. Published copy, also available at http://www.maths.warwick.ac.uk/gt/GTVol4/paper2.abs.htm

Decision problems and profinite completions of groups

We consider pairs of finitely presented, residually finite groups P\hookrightarrow\G for which the induced map of profinite completions \hat P\to \hat\G is an isomorphism. We prove that there is no algorithm that, given an arbitrary such pair, can determine whether or not $P$ is isomorphic to \G. We construct pairs for which the conjugacy problem in \G can be solved in quadratic time but the conjugacy problem in $P$ is unsolvable. Let $\mathcal J$ be the class of super-perfect groups that have a compact classifying space and no proper subgroups of finite index. We prove that there does not exist an algorithm that, given a finite presentation of a group \G and a guarantee that \G\in\mathcal J, can determine whether or not \G\cong\{1\}. We construct a finitely presented acyclic group \H and an integer $k$ such that there is no algorithm that can determine which $k$-generator subgroups of \H are perfect

The strong profinite genus of a finitely presented group can be infinite

We construct the first example of a finitely-presented, residually-finite group that contains an infinite sequence of non-isomorphic finitely-presented subgroups such that each of the inclusion maps induces an isomorphism of profinite completions.Comment: 10 pages, no figures. Final version to appear in Journal of the European Math. So

Limits of (certain) CAT(0) groups, I: Compactification

The purpose of this paper is to investigate torsion-free groups which act properly and cocompactly on CAT(0) metric spaces which have isolated flats, as defined by Hruska. Our approach is to seek results analogous to those of Sela, Kharlampovich and Miasnikov for free groups and to those of Sela (and Rips and Sela) for torsion-free hyperbolic groups. This paper is the first in a series. In this paper we extract an R-tree from an asymptotic cone of certain CAT(0) spaces. This is analogous to a construction of Paulin, and allows a great deal of algebraic information to be inferred, most of which is left to future work.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol5/agt-5-52.abs.htm

The simplicial boundary of a CAT(0) cube complex

For a CAT(0) cube complex $\mathbf X$, we define a simplicial flag complex $\partial_\Delta\mathbf X$, called the \emph{simplicial boundary}, which is a natural setting for studying non-hyperbolic behavior of $\mathbf X$. We compare $\partial_\Delta\mathbf X$ to the Roller, visual, and Tits boundaries of $\mathbf X$ and give conditions under which the natural CAT(1) metric on $\partial_\Delta\mathbf X$ makes it (quasi)isometric to the Tits boundary. $\partial_\Delta\mathbf X$ allows us to interpolate between studying geodesic rays in $\mathbf X$ and the geometry of its \emph{contact graph} $\Gamma\mathbf X$, which is known to be quasi-isometric to a tree, and we characterize essential cube complexes for which the contact graph is bounded. Using related techniques, we study divergence of combinatorial geodesics in $\mathbf X$ using $\partial_\Delta\mathbf X$. Finally, we rephrase the rank-rigidity theorem of Caprace-Sageev in terms of group actions on $\Gamma\mathbf X$ and $\partial_\Delta\mathbf X$ and state characterizations of cubulated groups with linear divergence in terms of $\Gamma\mathbf X$ and $\partial_\Delta\mathbf X$.Comment: Lemma 3.18 was not stated correctly. This is fixed, and a minor adjustment to the beginning of the proof of Theorem 3.19 has been made as a result. Statements other than 3.18 do not need to change. I thank Abdul Zalloum for the correction. See also: arXiv:2004.01182 (this version differs from previous only by addition of the preceding link, at administrators' request

The Schur multiplier, profinite completions and decidability

We fix a finitely presented group $Q$ and consider short exact sequences $1\to N\to G\to Q\to 1$ with $G$ finitely generated. The inclusion $N\to G$ induces a morphism of profinite completions $\hat N\to \hat G$. We prove that this is an isomorphism for all $N$ and $G$ if and only if $Q$ is super-perfect and has no proper subgroups of finite index. We prove that there is no algorithm that, given a finitely presented, residually finite group $G$ and a finitely presentable subgroup $P\subset G$, can determine whether or not $\hat P\to\hat G$ is an isomorphism.Comment: 6 pages no figures. To appear in the Bulletin London Math So