Abstract:
The talk is on a joint work with M. Sapir. We introduce a concept of
tree-graded metric spaces and use it to show quasi-isometry invariance of
certain classes of relatively hyperbolic groups, to obtain a
characterization of relatively hyperbolic groups in terms of their
asymptotic cones, to find geometric properties of Cayley graphs of
relatively hyperbolic groups, and to construct (using also ideas of
Olshanskii, Erschler and Osin) the first example of a finitely generated
group with continuum non-quasi-isoimetric asymptotic cones. Note that by a
result of Kramer, Shelah, Tent, and Thomas, continuum is the maximal
possible number of different asymptotic cones provided the Continuum
Hypothesis is true.