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Abstract
We give an algorithm deciding the generalized power problem for word hyperbolic groups. Alonso, Brady, Cooper, Ferlini, Lustig, Mihalik, Shapiro, and Short showed that elements of infinite order in word hyperbolic groups induce quasigeodesic rays in the Cayley graph. We show that a pair of quasigeodesic rays induced by two elements of infinite order either never meet at a vertex or intersect infinitely many times, and we give an algorithm detecting which option occurs for a given pair of quasigeodesic rays. Since solutions to the generalized power problem correspond to points of intersection along these rays, this decides instances of the generalized power problem involving elements of infinite order. For finite order instances, we use the algorithm deciding the word problem in word hyperbolic groups finitely many times. We extend this result to obtain an algorithm deciding membership in the product of two cyclic submonoids of a word hyperbolic group. We also give an algorithm deciding membership in finitely generated submonoids of the free product of two finitely presented groups, provided there is an algorithm to decide membership in the rational subsets of each factor. This extends a result of K. A. Mihailova, who proved that the uniform generalized word problem is decidable in the free product of two groups if it is decidable in each factor. Since rational membership is known to be decidable for free groups, free abelian groups, virtually free groups, and virtually free abelian groups, our algorithm can be used to decide membership in finitely generated submonoids of a free product of groups with factors drawn from these classes of groups.
