By Matthias Aschenbrenner;Stefan Friedl

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

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

**Example text**

By the theorem, the nilpotency class of ω is an upper bound on the lower p-length (and hence the nilpotency class) of G. We denote by ann(ω) := {r ∈ R : rω = 0} 32 2. EMBEDDING THEOREMS FOR p-GROUPS the left annihilator of ω (a left ideal of R[G]). Note that if ω d+1 = 0 then ann(ω) contains ω d . If G is ﬁnite we deﬁne g ∈ R[G]. G := g∈G Note that Gg = g G = G for all g ∈ G, hence G is central in R[G]. It is easy to see that if G is ﬁnite, then ann(ω) = RG; cf. 2]. The following observation is standard (cf.

19. Suppose that for each n ≥ 1, the morphism φe restricts to an ∼ = isomorphism Ge,n − → Ge ,n , and Gv ∩ fe (Ge ) is a uniformly p-potent ﬁltration of fe (Ge ). Then Gv ∩ fe (Ge ) is a uniformly p-potent ﬁltration of fe (Ge ). Proof. Note that the hypothesis that Gv ∩ fe (Ge ) be uniformly p-potent is equivalent to the ﬁltration Ge = (fe )−1 (Gv ) of Ge being uniformly p-potent, and that it is enough to show that Ge = fe−1 (Gv ) is a uniformly p-potent ﬁltration of Ge . Now by assumption, φe induces isomorphisms Ge,n → Ge ,n , for each n ≥ 1.

Then ) be a compatible collection of normal φ−1 (H) := φ−1 v (Hφ(v) ) v∈V (Y ) is a compatible collection of normal subgroups of G. 3) (left to the reader). In the rest of this section let G be a normal ﬁltration of G . 17, and if G is central, a p-ﬁltration, or p-potent, then clearly φ−1 (G ) also has the respective property. 18. Suppose each φv is injective. Then (1) if G is separating, then φ−1 (G ) is separating; (2) if each φe is bijective and G separates the edge groups of G , then φ−1 (G ) separates the edge groups of G; (3) if each φe maps Ge isomorphically onto the ﬁrst group Gφ(e),1 in the ﬁltration Gφ(e) of Gφ(e) , and G separates the edge groups of G , then φ−1 (G ) separates the edge groups of G.