Groups acting on Cantor sets and the end structure of graphs

We describe a variation of the Bergman norm for the algebra of cuts of a connected graph admitting a cofinite group action. By a construction of Dunwoody, this enables us to obtain nested generating sets for invariant subalgebras. We describe a few applications, in particular, to convergence groups acting on Cantor sets. Under certain finiteness assumptions one can deduce that such actions are necessarily geometrically finite, and hence arise as the boundaries of relatively hyperbolic groups. Similar results have already been obtained by Gerasimov by other methods. One can also use these techniques to give an alternative approach to the Almost Stability Theorem of Dicks and Dunwoody

Embedding median algebras in products of trees

Abstract. We show that a metric median algebra satisfying certain conditions admits a bilipschitz embedding into a finite product of R-trees. This gives rise to a characterisation of closed connected subalgebras of finite products of complete R-trees up to bilipschitz equivalence. Spaces of this sort arise as asymptotic cones of coarse median spaces. This applies to a large class of finitely generated groups, via their Cayley graphs. We show that such groups satisfy the rapid decay property. We also recover the result of Behrstock, Drut¸u and Sapir, that the asymptotic cone of the mapping class group embeds in a finite product of R-trees

Convergence groups and configuration spaces

We give an account of convergence groups from the point of view of groups which act properly discontinuously on spaces of distinct triples. We give a proof of the equivalence of this characterisation with the dynamical definition of Gehring and Martin. We focus our attention on uniform convergence groups, i.e. those for which the action on the space of distinct triples is also cocompact, and explore some of their properties from a purely dynamical point of view. We show that the space of distinct unordered n-tuples in any continuum is connected. Moreover, the spaces of distinct ordered n-tuples in any metrisable continuum other than a circle or an arc is also connected

Splittings of finitely generated groups over two-ended subgroups

We describe a means of constructing splittings of a one-ended finitely generated group over two-ended subgroups, starting with a finite collection of codimension-one two-ended subgroups. In the case where all the two-ended subgroups have two-ended commensurators, we obtain an annulus theorem, and a form of JSJ splitting of Rips and Sela. The construction uses ideas from the work of Dunwoody, Sageev and Swenson. We use a particular kind of order structure which combines cyclic orders and treelike structures. In the special case of hyperbolic groups, this provides a link between combinarorial constructions, and constructions arising from the topological structure of the boundary. In this context, we recover the annulus theorem of Scott and Swarup. We also show that a one-ended finitely generated groups which contains an infinite-order element, and such that every infinite cyclic subgroup is (virtually) codimension-one is a virtual surface group