On the subgroups of semihyperbolic groups

Martin R. Bridson

Preprint 2000 (revised and expanded from June 1997 preprint), in Monographie de L'Enseignement Math\'ematique, vol 38 (2001), 85--111

If a group $G$ acts properly and cocompactly by isometries on a simply-connected space of non-positive curvature then one can say a great deal about the structure of $G$. Some of the most significant properties of the groups $G$ that admit these cocompact actions are inherited by subgroups $\G\subset G$, but many others are not. Thus, given a compact non-positively curved space $X$, we investigate the subgroups $\G\subset\pi_1X$ and ask under what circumstances one can deduce that $\G$ is the fundamental group of a {\em compact} non-positively curved space. For manifolds of dimension $\le 3$ and complexes of dimension $\le 2$ it is shown that the finite presentability of a subgroup is sufficient to guarantee the existence of a cocompact action. In higher dimensions one encounters more subtle obstructions, for example higher finiteness conditions, the complexity of decision problems in the subgroup, and the structure of centralizers. We prove some general results concerning decision problems and use them to investigate certain closed non-positively curved manifolds $M$ and subgroups $\G\subset \pi_1M$ that arise as the fundamental groups of closed aspherical submanifolds $N_\G\subset M$; these subgroups $\G$ enjoy properties reminiscent of semihyperbolic groups but they are not semihyperbolic.