Intersection configurations in free times free-abelian groups

Abstract

In the present paper, we analyze closer the non-Howsonicity of the family of free times free-abelian groups Fn × Z m . On one hand, we give an algorithm to decide, given k 2 finitely generated subgroups H 1 , . . . , H k Fn × Z m , whether the intersection H 1 ∩ • • • ∩ H k is again finitely generated and, in the affirmative case, compute a basis for it. On the second hand, we show that any k-intersection configuration is realizable in Fn × Z m , for n = 2 and m big enough.

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

“…The isomorphism (11) follows immediately from (6). The equalities (12) and ( 13) are consequences of the normality of M ρ −1 in F r (note that ( 12) also needs the assumed condition M ρ −1 = {1}).…”

“…According to Theorem 4.14, St((H 1 ∩ H 2 )π, {w 1 , w 2 }) Cay(Z/Z ⊕ Z/0Z, {(1, 0), (0, 1)}) Cay(Z, {0, 1}), which takes the form: Since Z is infinite, in Case 2 the intersection H 1 ∩ H 2 has infinite rank. After replacing the arcs reading w 1 and w 2 with the enriched paths reading x 6 t (6,6), (6,6) and yx 3 y −1 t (3,3),(0,0) , folding, and normalizing (w.r.t. the spanning tree having as cyclomatic arcs the thicker ones), and equalizing, we obtain a Stallings automaton for H 1 ∩ H 2 : The corresponding (infinite) basis for H 1 ∩ H 2 is {yx 3k y −1 x 6 yx −3k y −1 t (6,6) : k ∈ Z}.…”

“…After replacing the arcs reading w 1 and w 2 with the enriched paths reading x 6 t (6,6), (6,6) and yx 3 y −1 t (3,3),(0,0) , folding, and normalizing (w.r.t. the spanning tree having as cyclomatic arcs the thicker ones), and equalizing, we obtain a Stallings automaton for H 1 ∩ H 2 : The corresponding (infinite) basis for H 1 ∩ H 2 is {yx 3k y −1 x 6 yx −3k y −1 t (6,6) : k ∈ Z}.…”

“…The more involved theory for the general semidirect scenario is in progress and will be published in the near future; see [11]. The theory of intersections of free-times-abelian groups, in turn, have a natural continuation in [6] (where the possible configurations of multiple intersections are studied) and [7] (where the results in [6] are used to obtain groups which, in some sense, admit a structure of quotients as complicated as possible). So, we revisit the family of free-times-abelian groups (already considered by the same authors in [9]), now from a geometric point of view.…”

“…In Section 2 we introduce the family of free-times-abelian groups (G = F n × Z m ) together with some related terminology and notation. It turns out that this naive-looking family hides interesting features that translate into non-trivial problems; see [4,6,9,14,31,32].…”