All countable groups have cubic presentations

Abstract

Let G be a group with presentation such that each generator occurs at most countably many times in the set of relations. Then for all n 3. G is ,I-ic. In particular, for all n 2 3, countable groups have n-ic presentations. Neuwirth (I) calls a group n-ic if it admits a presentation in which each generator appears exactly n times in the set of relations; such a presentation is also called n-ic. He goes on to show that if G is the fundamental group of a closed n-manifold then G * F is n-ic for some free F. Hoare (unpublished; see (2, p. 1461) has shown for n = 3 that G itself is 3-ic (cubic), and Montesinos (3) extends thls result to 3-manifolds with boundary. Here we show that any group that admits a presentation in which each generator occurs at most countably many times in the set of relations is n-ic.