Hyperbolic groups of Fibonacci type and T(5) cyclically presented groups

Building on previous results concerning hyperbolicity of groups of Fibonacci type, we give an almost complete classification of the (non-elementary) hyperbolic groups within this class. We are unable to determine the hyperbolicity status of precisely two groups, namely the Gilbert-Howie groups H(9, 4), H(9, 7). We show that if H(9, 4) is torsion- free then it is not hyperbolic. We consider the class of T(5) cyclically presented groups and classify the (non-elementary) hyperbolic groups and show that the Tits alternative holds. (C) 2021 Elsevier Inc. All rights reserved.