The finiteness of the Reidemeister number of morphisms between almost-crystallographic groups

Abstract

We give a practical criterion to determine whether a given pair of morphisms between almost-crystallographic groups has a finite Reidemeister coincidence number. As an application, we determine all two-and three-dimensional almost-crystallographic groups that have the R∞ property. We also show that for a pair of continuous maps between oriented infra-nilmanifolds of equal dimension, the Nielsen coincidence number equals the Reidemeister coincidence number when the latter is finite.Mathematics Subject Classification (2010). Primary 20E45; Secondary 55M20, 20H15.

The Reidemeister Spectra of Low Dimensional Almost-Crystallographic Groups

“…This is also the case for most almost-crystallographic groups, e.g. in [5] it was shown that 207 of the 219 3-dimensional crystallographic groups and 15 of the 17 families of 3-dimensional (notcrystallographic) almost-crystallographic groups all have the R ∞ -property. Furthermore, in [4] it was shown that 4692 of the 4783 4-dimensional crystallographic groups admit the R ∞ property.…”

“…A first result allows us to easily determine whether an almostcrystallographic groups admits the R ∞ -property or not. Theorem 3.4 (see [5]). Let Γ be an n-dimensional almost-crystallographic group with holonomy group F ⊆ Aut(G) and ϕ = ξ (d,D) ∈ Aut(Γ) (where we use the notation of theorem 3.1).…”

“…Every almost-crystallographic group of dimension 1 or 2 is crystallographic. In [5] it was determined which 3-dimensional almost-crystallographic groups admit the R ∞ -property. We extend these results to dimension 4.…”

The R∞ property for nilpotent quotients of surface groups

“…We refer to the paper [12] for an overview of the families of groups which have been studied in this context until 2008. After that people have also studied almostcrystallographic groups [2,5], Baumslag-Solitar groups [9], braid groups [8], some classes of metabelian groups [10] and several classes of linear groups [26][27][28][29]. We would also like to point the reader to [14] where several aspects of the R ∞ property are discussed.…”

REIDEMEISTER SPECTRUM FOR METABELIAN GROUPS OF THE FORM Qn ⋊ ℤ AND ℤ[1/p]n ⋊ ℤ, p PRIME

“…Later, it was shown by various authors that the following groups belong to this class: (1) non-elementary Gromov hyperbolic groups [5,27]; (2) Baumslag-Solitar groups BS(m, n) = a, b | ba m b −1 = a n except for BS (1,1) [7]; (3) generalized Baumslag-Solitar groups, that is, finitely generated groups which act on a tree with all edge and vertex stabilizers being infinite cyclic [26]; (4) lamplighter groups Z n Z iff 2 | n or 3 | n [20]; (5) the solvable generalization Γ of BS (1, n) given by the short exact sequence 1 → Z 1 n → Γ → Z k → 1, as well as any group quasi-isometric to Γ [31], groups which are quasi-isometric to BS(1, n) [30] (while this property is not a quasi-isometry invariant); (6) the Thompson group F [2]; (7) saturated weakly branch groups including the Grigorchuk group and the Gupta-Sidki group [12]; (8) mapping class groups, symplectic groups and braids groups [8]; (9) relatively hyperbolic groups [6]; (10) some classes of finitely generated free nilpotent groups [19,28] and some classes of finitely generated free solvable groups [25]; (11) some classes of crystallographic groups [3].…”