On the generators of the edge stabilizers of groups acting on trees

Abstract EN

In this paper we prove that if G is a group acting on a tree X such that the set of vertices of the quotient graph X/G is finite, then: (j) if the stabilizer Gx of each edge x of X is finitely generated, then G is finitely generated if and only if X/G is finite and the stabilizer Gy of each vertex v of X is finitely generated; (») if the stabilizer Gv of each vertex v of X is finitely presented then G is finitely presented if and only if X/G is finite and the stabilizer Gi of each edge x of X is finitely generated.