We consider acyclic groups of low dimension. To indicate our results simply, let G′ be the nontrivial perfect commutator subgroup of a finitely presentable group G. Then def(G) ≤ 1. When def(G) = 1, G′ is acyclic provided that it has no integral homology in dimensions above 2 (a sufficient condition for this is that G′ be finitely generated); moreover, G/G′ is then Z or Z. Natural examples are ...

by the mordell-weil theorem‎, ‎the group of rational points on an elliptic curve over a number field is a finitely generated abelian group‎. ‎there is no known algorithm for finding the rank of this group‎. ‎this paper computes the rank of the family $ e_p:y^2=x^3-3px $ of elliptic curves‎, ‎where p is a prime‎.

We show that the membership problem in a finitely generated submonoid of a graph group (also called a right-angled Artin group or a free partially commutative group) is decidable if and only if the independence graph (commutation graph) is a transitive forest. As a consequence we obtain the first example of a finitely presented group with a decidable generalized word problem that does not have ...

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 gro...