Boundaries/Section11 - Problems in Geometric Group Theory

11 Asymptotic cones

A geodesic metric space $X$ (e.g. Cayley graph of a finitely-generated group) is Gromov-hyperbolic if and only if all asymptotic cones of $X$ are trees. There are examples of finitely generated (such groups are never finitely-presented) groups $G$ so that some asymptotic cones of $G$ are trees but $G$ 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 $G$ admits an infinite presentation

$$G=< x_1,...,x_n| R_1, R_2, R_3,...>$$

so that each $G_k=<x_1,...,x_n| R_1,...R_k>$ 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 $Out(G)$ is finite or $G$ 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 $G$ , all asymptotic cones of $G$ 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?

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