Algorithms in geometric group theory

This dissertation explores algorithmic problems and properties occurring in geometric group theory. Special attention is paid to connections with formal language theory. Some of the results obtained are as follows: We investigate the change in complexity of certain algorithms under passing to finite extensions. For example, it is shown that a polynomial time solution for the generalized word problem gives rise to a polynomial time solution in any finite extension. A purely language theoretic definition of word hyperbolicity is given: a group is hyperbolic if and only if its word problem is growing and terminating. An automata theoretic construction called the “balloon construction” is introduced. The balloon construction yields a method for computing the angle between quasiconvex subgroups whose join is quasiconvex, inside hyperbolic groups. We define a group theoretic property called “super local quasiconvexity” and use the balloon construction to show that virtually free groups satisfy it. It is also shown that if S is a rational subset of a finitely generated virtually free group or virtually abelian group then S satisfies: the complement of S is rational, the membership problem of S is solvable, and S is unambiguously rational.