Abstract
Our first result is a decomposition theorem for free groups relative to a set of elements. This enables us to formulate several algebraic conditions, some necessary and some sufficient, for various surfaces in the boundary of a 3-dimensional handlebody to be incompressible. Moreover, we show that there exists an algorithm to determine whether or not these algebraic conditions are met.Many of our algebraic ideas are similar to those of Shenitzer [3]. Conversations with Professor Roger Lyndon were helpful in the initial development of these results, and he reviewed an earlier version of this paper, suggesting Theorem 1 (iii) and its proof. Our notation and techniques are standard (cf. [1], [2]). A set X of elements in a finitely generated free group F is a basis if it is a minimal generating set, and X±l denotes the set of all elements in X, together wTith their inverses.
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