On the Intersection of Double Cosets in Free Groups, with an Application to Amalgamated Products

Algorithmic problems for free-abelian times free groups

“…This is not true for free groups F n with n ⩾ 2, but it is also a classical result that all these groups satisfy the Howson property: the intersection of two finitely generated subgroups is again finitely generated. This elementary property fails dramatically in Z m ×F n , when m ⩾ 1 and n ⩾ 2 (a very easy example reproduced below, already appears in [7] attributed to Moldavanski). Consequently, the algorithmic problem of computing intersections of finitely generated subgroups of Z m × F n (including the preliminary decision problem on whether such intersection is finitely generated or not) becomes considerably more involved than the corresponding problems in Z m (just consisting on a system of linear equations over the integers) or in F n (solved by using the pull-back technique for graphs).…”

“…Despite it could seem against intuition, the same result fails dramatically when replacing the free product by a direct product. And one can find an extremely simple counterexample for this, in the family of free-abelian times free groups; the following observation is folklore (it appears in [7] attributed to Moldavanski, and as the solution to exercise 23.8(3) in [4]). Proof.…”

Stallings automata for free-times-abelian groups: intersections and index

“…In [2], B. Baumslag extended Howson's result by showing that the free product of Howson groups is again Howson. However, the same is not true for direct products: Moldavanski (see [3]) already showed that, in F {x,y} × Z, the intersection of the easy looking subgroups xt, y and x, y is the normal closure of y in F {x,y} , which is not finitely generated; see Section 5.1 below for our geometric interpretation of this interesting example. Therefore, in this context the Subgroup Intersection Problem SIP(G) emerges as a natural and interesting question, specially the decision part (which trivializes in the free case).…”

“…After replacing the arcs reading w 1 (resp., w 2 ) by an enriched path reading x 6 t (2,0),(0,3) (resp., yx 3 y −1 t (1,0),(0,0) ), folding, and normalizing w.r.t. a spanning tree T (whose cyclomatic arcs are drawn thicker), the automaton in Figure 13 becomes: This provides the basis { yx 3 y −1 x 6 t (3,0) , yx 6 y −1 x 6 yx −3 y −1 t (3,0) , yx 9 y −1 x 6 yx −6 y −1 t (3,0) , yx 12 y −1 x 6 yx −9 y −1 t (3,0) , yx 15 y −1 x 6 yx −12 y −1 t (3,0) , yx 18 y −1 t (6,−6) , x 6 yx −12 y −1 t (−3,6) } for the intersection…”

Intersection configurations in free times free-abelian groups

“…As a generalization of Howson's result, B. Baumslag established in [1] the preservation of the Howson property under free products: if G 1 and G 2 satisfy the Howson property then so does G 1 * G 2 . However, the same statement fails dramatically when replacing the free product by a direct product: the following observation is folklore (it appears in [3] attributed to Moldavanski, and as the solution to Exercise 23.8(3) in [2]).…”