Degrees of compression and inertia for free-abelian times free groups

Abstract

We introduce the concepts of degree of inertia, di(H), and degree of compression, dc(H), of a finitely generated subgroup H of a given group G. For the case of direct products of free-abelian and free groups, we compute the degree of compression and give an upper bound for the degree of inertia.Observe that (directly from the definition and using induction) inert subgroups are closed under finite intersections. Also, inert subgroups are compressed, while the other implication is not true in general: Example 1.2. Consider the direct product of the Klein bottle group with the group of integers, say G = a, b | bab −1 a × c | , and its subgroup H = a, b 2 , c ≃ Z 3 . By Corollary 4.3 and Proposition 4.4 from [20], H is compressed but not inert in G.

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

“…In Section 1 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 [6,9,26,27].…”