On the intersection of double cosets in free groups, with an application to amalgamated products
It is well known and elementary to prove that if \(H\), \(K\) are subgroups of a group \(G\), then the intersection \(Hg_1\cap Kg_2\) of any two cosets of \(H\) and \(K\) respectively is either empty or a single (right) coset of \(H\cap K\). However if one considers double cosets i.e. \(Hg_1U\) and \(Kg_2U\) where \(U\) is a third subgroup of \(G\) then their intersection \(Hg_1U\cap Kg_2U\) may contain more than one and even infinitely many \((H\cap K,U)\)-double cosets. In this paper the authors prove that if the group is free and the subgroup \(U\) is cyclic then the number of double cosets in the intersection is finite and bounded by certain invariants. More precisely one of the main results states: Let \(F\) be a free group, and let \(H\), \(K\) and \(U\) be subgroups of finite, nonzero ranks \(h\), \(k\), and \(1\), respectively. Let \(u\) denote a generator of \(U\), and write \(| u|\) for the length of \(u\) in terms of any set \(X\) of free generators of \(F\). Then each intersection \(Hg_1U\cap Kg_2U\), \(g_1,g_2\in F\), contains at most \[ [8| u|(h-1)+3\mu][8| u|(k-1)+3\mu] \] double cosets of the form \((H\cap K)gU\), where \(\mu\) is the largest integer such that \(w^\mu=u\) for some \(w\in F\). The second Theorem is an application of the above to amalgamated products \(G=A*_U B\) and is stated in terms of Karrass-Solitar rank and it is difficult to state here. However we can give the Corollary to this Theorem: Consider \(G=A*_U B\) where \(A\) is free and \(U\) is maximal cyclic in \(A\). If \(H,K\leq G\) are finitely generated subgroups intersecting all conjugates of \(A\) and \(B\) trivially, then \(H\), \(K\) and \(H\cap K\) are free, and \[ \text{rank}(H\cap K)\leq 8m^2n^2+\dots, \] where \(m\) and \(n\) are the ranks of \(H\) and \(K\). The authors remark that much of the paper consists of a re-ordering, emendation, and reproving in a more natural context of the main results of a previous paper of the first author [in Proc. 2nd internat. Conf. Theory of Groups, Canberra 1973, Lect. Notes Math. 372, 165-187 (1974; Zbl 0294.20043)].
scientific article; zbMATH DE number 1247572