Panos Papasoglu

Isodiametric and Isoperimetric Inequalities for Complexes and Groups

It is shown that D. Cohens inequality bounding the isoperimetric function of a group by the double exponential of its isodiametric function is valid in the more general context of locally finite simply connected complexes. It is shown that in this context this bound is ‘best possible’. Also studied are second-dimensional isoperimetric functions for groups and complexes. It is shown that the second-dimensional isoperimetric function of a group is bounded by a recursive function. By a similar argument it is shown that the area distortion of a finitely presented subgroup of a finitely presented group is recursive. Cohens inequality is extended to second-dimensional isoperimetric and isodiametric functions of 2-connected simplicial complexes.

From continua to ℝ–trees

We show how to associate an R-tree to the set of cut points of a continuum. If X is a continuum without cut points we show how to associate an R-tree to the set of cut pairs of X.

Codimension one subgroups and boundaries of hyperbolic groups

We construct hyperbolic groups with the following properties: The boundary of the group has big dimension, it is separated by a Cantor set and the group does not split. This shows that Bowditchs theorem that characterizes splittings of hyperbolic groups over 2-ended groups in terms of the boundary can not be extended to splittings over more complicated subgroups.

Growth and Isoperimetric Profile of Planar Graphs

In this section we review a joint work with Panos Papasoglu, see [BP11], in which the following is proved:

Splittings and the asymptotic topology of the lamplighter group

It is known that splittings of finitely presented groups over 2-ended groups can be characterized geometrically. We show that this characterization does not extend to all finitely generated groups. Answering a question of Kleiner we show that the Cayley graph of the lamplighter group is coarsely separated by quasi-circles.