Reidemeister spectra for solvmanifolds in low dimensions

In this paper we study the number of twisted conjugacy classes (the Reidemeister number) for automorphisms of crystallographic groups. We present two main algorithms for crystallographic groups whose holonomy group has finite normaliser in GLn(Z). The first algorithm calculates whether a group has the R∞-property; the second calculates the Reidemeister spectrum. We apply these algorithms to crystallographic groups up to dimension 6. * Supported by long term structural funding -Methusalem grant of the Flemish Government. Reidemeister numbers and spectrumIn this section we introduce basic notions concerning the Reidemeister number. For a general reference on Reidemeister numbers and its connection with fixed point theory, we refer the reader to [14].

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“…3/2/1/1/2 This is a Bieberbach group. In [6,Proposition 4.8], it was shown that Spec R (Γ) = 2N ∪ {∞}.…”

Section: Det(d) = 1 Then the Formula Becomesmentioning

confidence: 99%

The Reidemeister spectra of low dimensional crystallographic groups

2019

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“…The second equality follows from repeated applications of [8,Lemma 2.1]. For the fourth equality, see Remark 4.5.…”

Section: 1mentioning

confidence: 96%

“…Note that, although N (f ) = 0, the map f has R(f ) = ∞. Indeed, the subgroup G := z is invariant under ϕ and Π/G ∼ = Z, so we can use [8,Lemma 2.1] to compute that so in particular, T (v)X almost Φ-commutes with U . Hence…”

Section: 1mentioning

confidence: 99%

An Averaging Formula for Nielsen numbers on Infra-Solvmanifolds

Dekimpe¹,

Bussche²

2020

Preprint

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Until now only for special classes of infra-solvmanifolds, namely infra-nilmanifolds and infra-solvmanifolds of type (R), there was a formula available for computing the Nielsen number of a self-map on those manifolds. In this paper, we provide a general averaging formula which works for all selfmaps on all possible infra-solvmanifolds and which reduces to the old formulas in the case of infra-nilmanifolds or infra-solvmanifolds of type (R). Moreover, when viewing an infra-solvmanifold as a polynomial manifold, we recall that any map is homotopic to a polynomial map and we show how our formula can be translated in terms of the Jacobian of that polynomial map.

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“…Both of these semidirect products were studied in [6,Proposition 5.23], their Reidemeister spectra are respectively 4N ∪ {∞} and 8N ∪ {∞}.…”

Section: Spectra Of 4d Almost-bieberbach Groupsmentioning

confidence: 99%