A Note on Group Extensions and Proper 3-Realizability

Cárdenas, M.; Lasheras, Francisco F.; Quintero, A.; Roy, Ranja

doi:10.1007/s00009-016-0686-8

Cited by 4 publications

(3 citation statements)

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“…It is worth mentioning that each of these proper 2-equivalence classes contains non 3manifold groups as well, see Remark 5.23 below. Theorem 5.1 above together with [16,Thm. 1.2]) yield the following…”

Section: The Particular Case Of Properly 3-realizable Groupsmentioning

confidence: 86%

“…group extensions) are 1-ended and semistable at infinity [39] and have pro-isomorphic fundamental pro-groups, by Proposition 2.9. It is worth mentioning that they are all of telescopic type at infinity (see §5), i.e., their fundamental pro-groups are always pro-free and pro-(finitely generated) [18,16]; in fact, they are all proper 2-equivalent to one of the following groups Z × Z × Z, Z × Z or F 2 × Z, as we will show in §5.…”

Section: Introductionmentioning

confidence: 92%

“…It is worth mentioning that direct products and ascending HNN-extensions of infinite finitely presented groups have been shown to be of telescopic type at infinity [18,35]. In fact, any extension of an infinite finitely presented group by another infinite finitely presented group is of telescopic type at infinity [16]. One-relator groups are also of telescopic type at each end [15] (see also [36]).…”

Section: The Particular Case Of Properly 3-realizable Groupsmentioning

confidence: 99%

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A topological equivalence relation for finitely presented groups

Cárdenas

Lasheras

Quintero

et al. 2020

Journal of Pure and Applied Algebra

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In this paper, we consider an equivalence relation within the class of finitely presented discrete groups attending to their asymptotic topology rather than their asymptotic geometry. More precisely, we say that two finitely presented groups G and H are "proper 2-equivalent" if there exist (equivalently, for all) finite 2-dimensional CW-complexes X and Y , with π 1 (X) ∼ = G and π 1 (Y ) ∼ = H, so that their universal covers X and Y are proper 2-equivalent. It follows that this relation is coarser than the quasi-isometry relation. We point out that finitely presented groups which are 1-ended and semistable at infinity are classified, up to proper 2-equivalence, by their fundamental progroup, and we study the behaviour of this relation with respect to some of the main constructions in combinatorial group theory. A (finer) similar equivalence relation may also be considered for groups of type Fn, n ≥ 3, which captures more of the large-scale topology of the group. Finally, we pay special attention to the class of those groups G which admit a finite 2-dimensional CW-complex X with π 1 (X) ∼ = G and whose universal cover X has the proper homotopy type of a 3-manifold. We show that if such a group G is 1-ended and semistable at infinity then it is proper 2-equivalent to either Z × Z × Z, Z × Z or F 2 × Z (here, F 2 is the free group on two generators). As it turns out, this applies in particular to any group G fitting as the middle term of a short exact sequence of infinite finitely presented groups, thus classifying such group extensions up to proper 2-equivalence.Date: December 1, 2019.

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Section: The Particular Case Of Properly 3-realizable Groupsmentioning

confidence: 86%