2,039 research outputs found
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 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 is unsolvable.
Let 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 such
that there is no algorithm that can determine which -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 , we define a simplicial flag complex
, called the \emph{simplicial boundary}, which is a
natural setting for studying non-hyperbolic behavior of . We compare
to the Roller, visual, and Tits boundaries of
and give conditions under which the natural CAT(1) metric on
makes it (quasi)isometric to the Tits boundary.
allows us to interpolate between studying geodesic
rays in and the geometry of its \emph{contact graph} , 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 using
. Finally, we rephrase the rank-rigidity theorem of
Caprace-Sageev in terms of group actions on and
and state characterizations of cubulated groups with
linear divergence in terms of and .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 and consider short exact sequences
with finitely generated. The inclusion
induces a morphism of profinite completions . We prove that
this is an isomorphism for all and if and only if 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 and a finitely presentable subgroup ,
can determine whether or not is an isomorphism.Comment: 6 pages no figures. To appear in the Bulletin London Math So
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