Unsolvable Problems About Small Cancellation and Word Hyperbolic Groups

G. Baumslag, C. F. Miller, H. Short
1994 Bulletin of the London Mathematical Society  
We apply a construction of Rips to show that a number of algorithmic problems concerning certain small cancellation groups and, in particular, word hyperbolic groups, are recursively unsolvable. Given any integer k > 2, there is no algorithm to determine whether or not any small cancellation group can be generated by either two elements or more than k elements. There is a small cancellation group E such that there is no algorithm to determine whether or not any finitely generated subgroup of E
more » ... s all of E, or is finitely presented, or has a finitely generated second integral homology group.
doi:10.1112/blms/26.1.97 fatcat:5hettjmfq5aqrmxg6wtmedftgy