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

摘要

We discuss a problem posed by Gersten: Is every automatic group which doesnot contain Z+Z subgroup, hyperbolic? To study this question, we define thenotion of "n-tracks of length n", which is a structure like Z+Z, and prove itsexistence in the non-hyperbolic automatic groups with mild conditions. As anapplication, we show that if a group acts effectively, cellularly, properlydiscontinuously and cocompactly on a CAT(0) cube complex and its quotient is"weakly special", then the above question is answered affirmatively.