By Matthias Aschenbrenner;Stefan Friedl

Given a first-rate $p$, a gaggle 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 famous that finitely generated linear teams over fields of attribute 0 are almost residually $p$ for all yet finitely many $p$. specifically, primary 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 end up a typical generalisation of those effects: each $3$-manifold staff is nearly residually $p$ for all yet finitely many $p$. this provides facts for the conjecture (Thurston) that basic teams of $3$-manifolds are linear teams

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En ) is the geodesic in Y from v to v0 . In particular, ϕv0 = idGv0 , and ϕt(e) ◦ fe = ϕt(e) ◦ fe for every edge e of Y . 1. 5. There exists a unique morphism ϕ : G → Gv0 such that for each vertex v of Y , the diagram ϕ / Gv GO 0 ④= ④ ④ ④ ④④ ϕ ④④ v Gv commutes, where Gv → G is the natural morphism. 5 is implicit in [CM05]. A ﬁnite group Γ is homogeneous (in the sense of model theory) if every isomorphism between subgroups of Γ is induced by an automorphism of Γ. Finite homogeneous groups have been completely classiﬁed [CF00]; for example, given a prime p, the group of the form Z/pk Z ⊕ · · · ⊕ Z/pk Z (for some k) are homogeneous, and if p is odd, then these are the only homogeneous p-groups.

Suppose that G admits a ϕ-invariant chief ﬁltration G = {Gn } such that ϕ(a) ≡ a mod Gn+1 for all n and all a ∈ A ∩ Gn . Then G embeds into a p-group H such that ϕ extends to an inner automorphism of H and G is induced by a chief ﬁltration of H. This fact easily yields the following corollary, which generalizes a well-known characterization of unipotent subgroups of the general linear group over a ﬁeld of characteristic p; cf. 14. The following are equivalent: (1) There exists a p-group H which contains G as a subgroup such that each ϕi extends to an inner automorphism of H.

Suppose H is a p-group. Then W = R H is nilpotent, and the nilpotency class, lower p-length, and lower dimensional p-length of W all equal p n dimFp (Dnp (H)/Dn+1 (H)). 10. Let A be a group containing R as a central subgroup, let θ : A → H be a group morphism to a p-group H (not necessarily surjective) with kernel R, and let α : A → W = R H be a corresponding standard embedding. Then p α(R) = ω d = γd+1 (W ) ∩ R[H] = RH where d is the nilpotency class of ω = ω(R[H]). ∗ Proof. Let θ ∗ be a countermap to θ such that α(a) = (θ(a), faθ ) for all a ∈ A.