12 research outputs found
The fully residually F quotients of F*<x,y>
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.Comment: 64 pages, 2 figures. Following recommendations from a referee, the
paper has been completely reorganized and many small mistakes have been
corrected. There were also a few gaps in the earlier version of the paper
that have been fixed. In particular much of the content of Section 8 in the
previous version had to be replaced. This paper is to appear in Groups. Geom.
Dy
On the quasi-isometric rigidity of graphs of surface groups
Let 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 quasi-isometric to is
abstractly commensurable with .Comment: 54 pages, 10 figures, comments welcom
Panel collapse and its applications
We describe a procedure called panel collapse for replacing a CAT(0) cube
complex by a "lower complexity" CAT(0) cube complex
whenever contains a codimension- hyperplane that is extremal in one
of the codimension- hyperplanes containing it. Although is
not in general a subcomplex of , it is a subspace consisting of a
subcomplex together with some cubes that sit inside "diagonally". The
hyperplanes of extend to hyperplanes of . Applying this
procedure, we prove: if a group acts cocompactly on a CAT(0) cube complex
, then there is a CAT(0) cube complex so that acts
cocompactly on and for each hyperplane of , the stabiliser
in of acts on 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 on the
free splitting complex. Finally, we apply panel collapse to a conjecture of
Kropholler, obtaining a short proof under a natural extra hypothesis.Comment: Revised according to referee comments. This version accepted in
"Groups, Geometry, and Dynamics
Detecting geometric splittings in finitely presented groups
We present an algorithm which given a presentation of a group without
2-torsion, a solution to the word problem with respect to this presentation,
and an acylindricity constant , outputs a collection of tracks in an
appropriate presentation complex. We give two applications: the first is an
algorithm which decides if admits an essential free decomposition, the
second is an algorithm which; if is relatively hyperbolic; decides if it
admits an essential elementary splitting.Comment: This is a rewritten version of the paper "Finding tracks in
2-complexes". The statements of the main theorems are unchanged and the proof
is essentially the same, but the presentation has been substantially
improved. To appear in Transactions of the American Mathematical Societ
Multipass automata and group word problems
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