Abstract

A well-known open problem in group theory is the existence of a group with cohomological dimension 2 and geometric dimension 3, known as the Eilenberg-Ganea problem. For relative versions of this problem there are examples of groups with relative cohomological dimension 2 and relative geometric dimension 3 due to the work of several authors. We call the aforementioned groups relative Eilenberg-Ganea groups. On the other hand, in geometric group theory, it is usual to try to classify groups up to quasi-isometry. In this talk we will explore roughly how many non quasi-isometric groups are among the class of relative Eilenberg-Ganea groups. Explicitly, using a quasi-isometry invariant introduced by Bowditch, called the Taut spectrum, we will sketch how to prove that there are continuously many non-quasi-isometric groups with proper (resp. virtually cyclic) cohomological dimension 2 and proper (resp. virtually cyclic) geometric dimension 3. This is joint work with Eduardo Martínez-Pedroza.