Non-hyperbolic automatic groups and groups acting on CAT(0) cube complexes

This note is a brief summary of my talk that I gave at RIMS workshop “Complex Analysis and Topology of Discrete Groups and Hyperbolic Spaces.” See [4] for detail. If a group G has a finite K(G, 1) and does not contain any Baumslag–Solitar groups, is G hyperbolic? (See [1].) This is one of the most famous questions on hyperbolic groups. Probably, many people expect that the answer is negative, and it would be better to restrict our attention to some good class of groups. In this talk, we consider automatic groups. If an automatic group G does not contain any Z + Z subgroups, is G hyperbolic? Our problem is listed in [5] and attributed to Gersten. Note that, if the group is the fundamental group of a closed 3-manifold, our question corresponds to the so-called “weak hyperbolization” of 3-manifolds. In this talk, we define the notion of “n-tracks of length n”, which suggests a clue of the existence of Z + Z subgroup, and show its existence in every non-hyperbolic automatic groups with mild conditions. As an application, we show that if a group acts freely, cellularly, properly discontinuously and cocompactly on a CAT(0) cube complex and its quotient is “weakly special”, then the above question is answered affirmatively. See [4] for detail.