157 research outputs found
Further topics in connectivity
Continuing the study of connectivity, initiated in §4.1 of the Handbook, we survey here some (sufficient) conditions under which a graph or digraph has a given connectivity or edge-connectivity. First, we describe results concerning maximal (vertex- or edge-) connectivity. Next, we deal with conditions for having (usually lower) bounds for the connectivity parameters. Finally, some other general connectivity measures, such as one instance of the so-called “conditional connectivity,” are considered.
For unexplained terminology concerning connectivity, see §4.1.Peer ReviewedPostprint (published version
Finiteness properties of cubulated groups
We give a generalized and self-contained account of Haglund-Paulin's
wallspaces and Sageev's construction of the CAT(0) cube complex dual to a
wallspace. We examine criteria on a wallspace leading to finiteness properties
of its dual cube complex. Our discussion is aimed at readers wishing to apply
these methods to produce actions of groups on cube complexes and understand
their nature. We develop the wallspace ideas in a level of generality that
facilitates their application.
Our main result describes the structure of dual cube complexes arising from
relatively hyperbolic groups. Let H_1,...,H_s be relatively quasiconvex
codimension-1 subgroups of a group G that is hyperbolic relative to
P_1,...,P_r. We prove that G acts relatively cocompactly on the associated dual
CAT(0) cube complex C. This generalizes Sageev's result that C is cocompact
when G is hyperbolic. When P_1,...,P_r are abelian, we show that the dual
CAT(0) cube complex C has a G-cocompact CAT(0) truncation.Comment: 58 pages, 12 figures. Version 3: Revisions and slightly improved
results in Sections 7 and 8. Several theorem numbers have changed from the
previous versio
- …