We completely describe the finitely generated pro-$p$ subgroups of the profinite completion of the fundamental group of an arbitrary $3$-manifold. We also prove a pro-$p$ analogue of the main theorem of Bass--Serre theory for finitely generated pro-$p$ groups.
We study 3-dimensional Poincare duality pro-$p$ groups in the spirit of the work by Robert Bieri and Jonathan Hillmann, and show that if such a pro-$p$ group $G$ has a nontrivial finitely presented subnormal subgroup of infinite index, then either th
e subgroup is cyclic and normal, or the subgroup is cyclic and the group is polycyclic, or the subgroup is Demushkin and normal in an open subgroup of $G$. Also, we describe the centralizers of finitely generated subgroups of 3-dimensional Poincare duality pro-$p$ groups.
The profinite completion of the fundamental group of a closed, orientable $3$-manifold determines the Kneser--Milnor decomposition. If $M$ is irreducible, then the profinite completion determines the Jaco--Shalen--Johannson decomposition of $M$.
We study the subgroup structure of the etale fundamental group $Pi$ of a projective curve over an algebraically closed field of characteristic 0. We obtain an analog of the diamond theorem for $Pi$. As a consequence we show that most normal subgroups
of infinite index are semi-free. In particular every proper open subgroup of a normal subgroup of infinite index is semi-free.
Let $w$ be a multilinear commutator word. In the present paper we describe recent results that show that if $G$ is a profinite group in which all $w$-values are contained in a union of finitely (or in some cases countably) many subgroups with a presc
ribed property, then the verbal subgroup $w(G)$ has the same property as well. In particular, we show this in the case where the subgroups are periodic or of finite rank.
We consider the question of which right-angled Artin groups contain closed hyperbolic surface subgroups. It is known that a right-angled Artin group $A(K)$ has such a subgroup if its defining graph $K$ contains an $n$-hole (i.e. an induced cycle of l
ength $n$) with $ngeq 5$. We construct another eight forbidden graphs and show that every graph $K$ on $le 8$ vertices either contains one of our examples, or contains a hole of length $ge 5$, or has the property that $A(K)$ does not contain hyperbolic closed surface subgroups. We also provide several sufficient conditions for a RAAG to contain no hyperbolic surface subgroups. We prove that for one of these forbidden subgraphs $P_2(6)$, the right angled Artin group $A(P_2(6))$ is a subgroup of a (right angled Artin) diagram group. Thus we show that a diagram group can contain a non-free hyperbolic subgroup answering a question of Guba and Sapir. We also show that fundamental groups of non-orientable surfaces can be subgroups of diagram groups. Thus the first integral homology of a subgroup of a diagram group can have torsion (all homology groups of all diagram groups are free Abelian by a result of Guba and Sapir).