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

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**Example text**

Then N is a normal subgroup of G with G/N satisfying P, v v∈V (Y ) and the natural morphism ψ : G → G/N is injective on each Gv . 6, Lemma 8]. ” Remark. The proof of the implications “(3) ⇒ (1) ⇒ (2)” did not use the ﬁniteness of the Gv , and the proof of “(2) ⇒ (3) ⇒ (1)” is valid without assuming that free groups are residually P. 4. Let H be a normal subgroup of G such that G/H satisﬁes P. Then π1 (G/H) is residually P, where H = {H ∩ Gv }v∈V (Y ) . Proof. 3 (the proof of which didn’t need ﬁniteness of the Gv /Hv ).

11. Let {Hn } be a ﬁltration of H. Then Hn B = {hb : h ∈ Hn , b ∈ B} deﬁnes a ﬁltration of G = B H. If {Hn } is normal, then so is {Hn B}, and for every m < n the natural morphism Hm /Hn → Hm B/Hn B is an isomorphism. ) Let now {Gn } be a central ﬁltration of G, and put Bn := B ∩ Gn for each n. Note that [G, B] ≤ B, hence [G, Bn ] ≤ Bn+1 for each n. Assume also that a central ﬁnite-length ﬁltration H = H 1 ≥ H 2 ≥ · · · ≥ Hm ≥ 1 of H is given. We can then combine the central ﬁltration {Bn } of B with the central ﬁltration {Hn B} of G to a central ﬁltration G = H 1 B ≥ · · · ≥ Hm B ≥ B = B 1 ≥ B 2 ≥ · · · ≥ B n ≥ · · · of G.

Matching up these components in pairs and connecting them via n copies of Xe , the resulting complex X has the desired property. Now assume Y has two distinct vertices v1 and v2 such that t(e) = v1 and (fe (Xe )), for i = 1, 2. We t(e) = v2 . Let ni be the number of components of πv−1 i get the cover X → X by taking lcm(n1 , n2 )/n1 copies of Xv1 and lcm(n1 , n2 )/n2 copies of Xv2 and by connecting each copy of a component of πv−1 (fe (Xe )) with a 1 copy of a component of πv−1 (f (X )) via a copy of X .

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

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