11 Asymptotic cones
A geodesic metric space
dvipng error! exitcode was 2 (signal 0), transscript follows:(e.g. Cayley graph of a finitely-generated group) is Gromov-hyperbolic if and only if all asymptotic cones of
dvipng error! exitcode was 2 (signal 0), transscript follows:are trees. There are examples of finitely generated (such groups are never finitely-presented) groups
dvipng error! exitcode was 2 (signal 0), transscript follows:so that some asymptotic cones of
dvipng error! exitcode was 2 (signal 0), transscript follows:are trees but
dvipng error! exitcode was 2 (signal 0), transscript follows:is not Gromov-hyperbolic, see MR1734187. Call such groups lacunary hyperbolic following Olshansky, Osin and Sapir see OOS. All such groups are limits of hyperbolic groups in the sense that
dvipng error! exitcode was 2 (signal 0), transscript follows:admits an infinite presentation
dvipng error! exitcode was 2 (signal 0), transscript follows:
so that each
dvipng error! exitcode was 2 (signal 0), transscript follows:is hyperbolic.
Problem 11.1 (Comments) [Misha Kapovich] Is there are meaningful structure theory for lacunary hyperbolic groups? Can one define a useful boundary for such groups? Is it true that either
dvipng error! exitcode was 2 (signal 0), transscript follows:is finite or
dvipng error! exitcode was 2 (signal 0), transscript follows:splits over a virtually cyclic subgroup?
Remark A counter-example to the last problem is known to M.~Sapir.
It is known that for each relatively hyperbolic group
dvipng error! exitcode was 2 (signal 0), transscript follows:, all asymptotic cones of
dvipng error! exitcode was 2 (signal 0), transscript follows:have cut points.
Problem 11.2 (Comments) [Cornelia Drutu] To what extent is the reverse implication true?
Remark Some counterexamples are known; for instance, the mapping class group and fundamental groups of graph manifolds are weakly relatively hyperbolic but not strongly.
Problem 11.3 (Comments) [Mario Bonk] The study of asymptotic cones has been non-analytic (they have been studied up to homeomorphism). What analytic tools could be developed?