The pq-condition for $3$-manifold groups

Abstract

We give an elementary, topological proof of the fact that any sugroup of order pq of a finite 3-manifold group is cyclic if p and q are dis- tinct odd primes. This condition, together with related results of Milnor and Reidemeister, implies that such a group acts orthogonally and without fixed points on some sphere. Our aim here is to understand the restrictions on finite groups that are fundamen- tal groups of 3-manifolds. Two necessary conditions are well known; namely that if G is a such a group, then its subgroups of order 2p, for p a prime, are cyclic(2)(the 2p-condition) and so are its subgroups of order p 2 (3)(4) (the p 2 -condition). The latter is equivalent to the statement that all abelian subgroups of G are cyclic. The p 2 -condition for 3-manifold groups is an immediate consequence of the fact that (Z/pZ) 2 does not have a balanced presentation. The above conditions together with the pq-condition characterise when solvable finite groups act orthogonally on some sphere (5) with fixed points. The pq-condition is the condition that every subgroup of order pq of G, where p and q are distinct, odd primes, is abelian (hence cyclic). Our main result is that finite fundamental groups of 3-manifolds satisfy this condition. This was previously known, but proofs in the literature (for example in (2)) involve using the fact that G has cohomology with periodicity 4 together with some deep group theory to restrict the class of finite fundamental groups, and then observing that this condition is satisfied. It seems desirable to deduce this result using a direct topological argument, especially given the fundamental role of this condition in characterising orthogonal free actions. The proof given here is elementary and topological. Theorem 0.1. Suppose the finite group G is the fundamental group of a 3-manifold M, and p and q are distinct, odd primes. Then every subgroup of G of order pq is cyclic.