By Matthias Aschenbrenner;Stefan Friedl

ISBN-10: 0821888013

ISBN-13: 9780821888018

Given a first-rate $p$, a bunch is named residually $p$ if the intersection of its $p$-power index basic subgroups is trivial. a gaggle is named nearly residually $p$ if it has a finite index subgroup that's residually $p$. it truly is recognized that finitely generated linear teams over fields of attribute 0 are nearly residually $p$ for all yet finitely many $p$. particularly, primary teams of hyperbolic $3$-manifolds are almost residually $p$. it's also famous that primary teams of $3$-manifolds are residually finite. during this paper the authors end up a typical generalisation of those effects: each $3$-manifold staff is nearly residually $p$ for all yet finitely many $p$. this offers proof for the conjecture (Thurston) that basic teams of $3$-manifolds are linear teams

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**Extra info for 3-manifold groups are virtually residually p**

**Sample text**

Ba73]. If each vertex group of G is a p-group, then G always has a free (but not necessarily normal) subgroup whose index is the least common multiple of the orders of the vertex groups (hence a power of p), cf. 6, Lemmas 8 and 10]; to have a free normal subgroup of p-power index is equivalent to G being residually p. 3. Suppose that each vertex group of G is a ﬁnite P-group. Then the following are equivalent: (1) G has a free normal subgroup N such that G/N has P. (2) G is residually P. (3) There is a morphism ψ : G → P to a P-group P such that ψ|Gv is injective for every v ∈ V (Y ).

An embedding of an amalgam G∪H|U into a group W is a pair (α, β) consisting of injective group morphisms α : G → W , β : H → W such that α|U = β|U . ) Such an embedding of G ∪ H|U into W is said to be strong if in addition α(G) ∩ β(H) = α(U ). The following is a variant of a theorem of Higman [Hig64]. 2, we are mostly interested in the addendum. 1. Let G ∪ H|U be an amalgam of p-groups. The following are equivalent: (1) G ∪ H|U is strongly embeddable into a p-group. (2) G ∪ H|U is embeddable into a p-group.

For the converse note that every chief ﬁltration of a p-group containing G as a subgroup induces a chief ﬁltration of G, and that every central ﬁltration is invariant under inner automorphisms. 2. 15. Suppose there exists a ﬁnite-length central ﬁltration G = {Gj } of G which is ϕi -invariant for each i and such that for each j there is a p-group Hj containing Lj (G) as a subgroup and for each i an extension of the partial automorphism Lj (ϕi ) of Lj (G) induced by ϕi to an inner automorphism of Hj .

### 3-manifold groups are virtually residually p by Matthias Aschenbrenner;Stefan Friedl

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