22 - On one-relator groups that are free products of two free groups with cyclic amalgamation

Summary

INTRODUCTION

Let G = < a₁, …, a_p, b₁, …, b_q ∣ wv = 1 >, 2 ≤ p, 2 ≤ q, where 1  w = w(a₁, …, a_p) is not a proper power nor a primitive element in the free group H = < a₁, …, a_p; > and 1  v = v(b₁, …, b_p) is not a proper power nor a primitive element in the free group H = < b₁, …, b_q; >. The group G is of great interest both for group theory and for topology (see and). We are concerned with the one-relator presentations of G and the solution of the isomorphism problem for G. In this paper we prove Theorem 3.19: If p = q = 2 and {x₁, …, x₄} is a generating system of G, then {x₁, …, x₄} is freely equivalent to a system {y₁, …, y₄) with {y₁, …, y₄} C H₁ ⋃ H₂. Moreover, for {x₁, …, x₄} there is a presentation of G with one defining relation. Also, G has only finitely many Nielsen equivalence classes of minimal generating systems, and we can decide algorithmically in finitely many steps whether an arbitrary one-relator group is or is not isomorphic to G.

This result stands in contrast to the corresponding results in and.