Centralisateurs d'éléments dans les PD(3)-paires

Abstract

On établit pour les centralisateurs dans une PD(3)-paire des résultats analogues à ceux connus pour les centralisateurs dans un groupe fondamental de variété de dimension 3. Comme dans le cas des groupes fondamentaux de variétés de dimension 3, la preuve de ces résultats repose sur une décomposition JSJ pour les PD(3)-paires obtenue à l'aide de la théorie des voisinages algébriques réguliers de Scott et Swarup.

On the Tits alternative forPD(3)groups

“…does not contain any Z ⊕ Z subgroup). This was generalised in F. Castel's thesis [Cas], where Scott-Swarup regular neighborhood theory ( [ScSw]) was used to show that an orientable P D(3) group admits a canonical JSJ splitting along free abelian groups of rank 2 with vertex groups isomorphic to either Seifert manifold groups or atoroidal P D(3) pairs in the sense of [Cas,Definition 1(2)]. This splitting is analogous to the one induced on the fundamental group of closed orientable aspherical 3-manifolds by their geometric decomposition.…”

Subnormality in $PD_3$-groups and $L^2$-Betti numbers

“…Since χ(G) is half the sum of the Euler characteristics of its boundary components, χ(T ) = 0. Since G 1 has the maximal condition on centralizers [4] (see also [12]) and has no non-trivial abelian normal subgroup, it splits over a subgroup commensurable with a conjugate of T [15, Theorem A1]. Since G does not split over T it must fix a vertex of the G 1 -tree corresponding to the splitting.…”

Around 3-manifold groups

“…This splitting as a graph of groups is analogous to the one induced on the fundamental group of a closed orientable aspherical 3manifold by the JSJ-splitting. See [DuSw00] and [Wa03,Theorem 10.8] or [Wa04, Theorem 4.2] for the case of finitely presentable PD(3) groups, and [Cas07] or [Kro90] and [Hil06] for the general case which avoids the finitely presentable assumption (cf. [Hil19] for more details).…”