39 research outputs found

### Computing the Maximum Slope Invariant in Tubular Groups

We show that the maximum slope invariant for tubular groups is easy to
calculate, and give an example of two tubular groups that are distinguishable
by their maximum slopes but not by edge pattern considerations or isoperimetric
function.Comment: 9 pages, 7 figure

### Quasi-isometry Invariants from Decorated Trees of Cylinders of Two-Ended JSJ Decompositions

We construct quasi-isometry invariants of a one-ended finitely presented
group by considering the tree of cylinders of a two-ended JSJ decomposition of
the group. When the group satisfies additional quasi-isometric rigidity
hypotheses we construct finer invariants by also considering relative amounts
of stretching across edges of the tree of cylinders.Comment: arXiv:1601.07147 now contains all of these results and much, much
mor

### Quasi-isometries need not induce homeomorphisms of contracting boundaries with the Gromov product topology

We consider a `contracting boundary' of a proper geodesic metric space
consisting of equivalence classes of geodesic rays that behave like geodesics
in a hyperbolic space. We topologize this set via the Gromov product, in
analogy to the topology of the boundary of a hyperbolic space. We show that
when the space is not hyperbolic, quasi-isometries do not necessarily give
homeomorphisms of this boundary. Continuity can fail even when the spaces are
required to be CAT(0). We show this by constructing an explicit example.Comment: 5 pages, to appear in Analysis and Geometry in Metric Space

### Quasi-isometries Between Tubular Groups

We give a method of constructing maps between tubular groups inductively
according to a set of strategies. This map will be a quasi-isometry exactly
when the set of strategies is consistent. Conversely, if there exists a
quasi-isometry between tubular groups, then there is a consistent set of
strategies for them.
There is an algorithm that will in finite time either produce a consistent
set of strategies or decide that such a set does not exist. Consequently, this
algorithm decides whether or not the groups are quasi-isometric.Comment: 44 pages, 11 figures. PDFLaTeX. Improved exposition and added some
auxiliary material to make the paper more self contained, per referee's
comment

### Morse subsets of CAT(0) spaces are strongly contracting

We prove that Morse subsets of CAT(0) spaces are strongly contracting. This
generalizes and simplifies a result of Sultan, who proved it for Morse
quasi-geodesics. Our proof goes through the recurrence characterization of
Morse subsets.Comment: 3 page

### Growth Tight Actions of Product Groups

A group action on a metric space is called growth tight if the exponential
growth rate of the group with respect to the induced pseudo-metric is strictly
greater than that of its quotients. A prototypical example is the action of a
free group on its Cayley graph with respect to a free generating set. More
generally, with Arzhantseva we have shown that group actions with strongly
contracting elements are growth tight.
Examples of non-growth tight actions are product groups acting on the $L^1$
products of Cayley graphs of the factors.
In this paper we consider actions of product groups on product spaces, where
each factor group acts with a strongly contracting element on its respective
factor space. We show that this action is growth tight with respect to the
$L^p$ metric on the product space, for all $1<p\leq \infty$. In particular, the
$L^\infty$ metric on a product of Cayley graphs corresponds to a word metric on
the product group. This gives the first examples of groups that are growth
tight with respect to an action on one of their Cayley graphs and non-growth
tight with respect to an action on another, answering a question of Grigorchuk
and de la Harpe.Comment: 13 pages v2 15 pages, minor changes, to appear in Groups, Geometry,
and Dynamic

### Quasi-isometries Between Groups with Two-Ended Splittings

We construct `structure invariants' of a one-ended, finitely presented group
that describe the way in which the factors of its JSJ decomposition over
two-ended subgroups fit together.
For groups satisfying two technical conditions, these invariants reduce the
problem of quasi-isometry classification of such groups to the problem of
relative quasi-isometry classification of the factors of their JSJ
decompositions. The first condition is that their JSJ decompositions have
two-ended cylinder stabilizers. The second is that every factor in their JSJ
decompositions is either `relatively rigid' or `hanging'. Hyperbolic groups
always satisfy the first condition, and it is an open question whether they
always satisfy the second.
The same methods also produce invariants that reduce the problem of
classification of one-ended hyperbolic groups up to homeomorphism of their
Gromov boundaries to the problem of classification of the factors of their JSJ
decompositions up to relative boundary homeomorphism type.Comment: 61pages, 6 figure

### A Metrizable Topology on the Contracting Boundary of a Group

The 'contracting boundary' of a proper geodesic metric space consists of
equivalence classes of geodesic rays that behave like rays in a hyperbolic
space. We introduce a geometrically relevant, quasi-isometry invariant topology
on the contracting boundary. When the space is the Cayley graph of a finitely
generated group we show that our new topology is metrizable.Comment: v1: 26 pages, 3 figures; v2: 44 pages, 6 figures, additional results;
v3: 46 pages, 7 figures, minor change