Virtual retraction and Howson’s theorem in pro-$p$ groups

The Hanna Neumann conjecture for Demushkin groups

“…We now combine the finiteness of S with Howson's theorem for Demushkin groups (see [32,Theorem 1.8]) into a single homological statement. Corollary 3.5.…”

“…Demushkin groups were also studied for their own sake, for instance, in [29] by Serre and in [17,18] by Labute. Their group theoretic properties continue to attract attention as can be seen from [5,12,13,14,15,31,32,33,35]. In particular, Howson's theorem for these groups has been obtained by Shusterman and Zalesskii in [32].…”

“…Their group theoretic properties continue to attract attention as can be seen from [5,12,13,14,15,31,32,33,35]. In particular, Howson's theorem for these groups has been obtained by Shusterman and Zalesskii in [32]. It is therefore very natural to ask whether the Hanna Neumann conjecture is true also for Demushkin groups.…”

Inertia of retracts in Demushkin groups

“…Finally, let H be a proper retract of an infinite Demushkin group G and take any closed and finitely generated subgroup K of G. By Proposition 3.2, we have β G 1 (I G/H ) = 0. Since all Demushkin groups possess Howson's property [9,Thm. 3.1], the intersection H ∩ K is a finitely generated subgroup, and therefore the relation gradient β K 1 (I K/(H∩K) ) is also defined.…”

Pro-p groups acting on trees with finitely many maximal vertex stabilizers up to conjugation

“…Free pro-p-groups are Howson, and the Howson property is preserved under free (pro-p) products, see [24,Thm 1.9]. In this section we investigate the preservation of Howson's property under various (free) constructions.…”