Abstract:Abstract. We discuss a problem posed by Gersten: Is every automatic group which does not contain Z + Z subgroup, hyperbolic? To study this question, we define the notion of "n-tracks of length n", which is a structure like Z + Z, and prove its existence in the non-hyperbolic automatic groups with mild conditions. As an application, we show that if a group acts effectively, cellularly, properly discontinuously and cocompactly on a CAT(0) cube complex and its quotient is "weakly special", then the above question… Show more
In this short note, we show that a group acting geometrically on a CAT(0) cube complex with virtually abelian hyperplane-stabilisers must decompose virtually as a free product of free abelian groups and surface groups.
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