Cabrer, Leonardo Manuel; Mundici, Daniele Classifying orbits of the affine group over the integers. (English) Zbl 1417.37114 Ergodic Theory Dyn. Syst. 37, No. 2, 440-453 (2017). Let \(\mathrm{GL}(n,\mathbb Z)\ltimes \mathbb Z^n\), \(n\in \mathbb N\), be the affine group over the integers. Let \(x=(x_1,\dots,x_n)\in \mathbb R^n\) and \(G_x\) be the subgroup of the additive group \(\mathbb R\) generated by \(1, x_1, \dots, x_n\). In the paper it is shown that \(x,y\in\mathbb R^n\) lie in the same \(\mathrm{GL}(n,\mathbb Z)\ltimes \mathbb Z^n\)-orbit if and only if \((G_x, c_x)=(G_y, c_y),\) where \(c_x\) is an integer associated to rational polyhedral geometry. Reviewer: Utkir A. Rozikov (Tashkent) Cited in 7 Documents MSC: 37C85 Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\) 20B07 General theory for infinite permutation groups Keywords:affine group; orbit; classification; additive group PDFBibTeX XMLCite \textit{L. M. Cabrer} and \textit{D. Mundici}, Ergodic Theory Dyn. Syst. 37, No. 2, 440--453 (2017; Zbl 1417.37114) Full Text: DOI arXiv References: [1] A.Bigard, K.Keimel and S.Wolfenstein. Groupes et Anneaux Réticulés(Lecture Notes in Mathematics, 608). Springer, Berlin, 1971. · Zbl 0384.06022 [2] F.Boca. An AF algebra associated with the Farey tessellation. Canad. J. Math.60 (2008), 975-1000.10.4153/CJM-2008-043-1 · Zbl 1158.46039 [3] J. S.Dani. Density properties of orbits under discrete groups. J. Indian Math. Soc. (N.S.)39 (1975), 189-218. · Zbl 0428.22006 [4] C.Eckhardt. A noncommutative Gauss map. Math. Scand.108 (2011), 233-250.10.7146/math.scand.a-15169 · Zbl 1228.46060 [5] E. G.Effros. Dimensions and C^∗ -Algebras(CBMS Regional Conference Series in Mathematics, 46). American Mathematical Society, Providence, RI, 1981.10.1090/cbms/046 [6] G.Ewald. Combinatorial Convexity and Algebraic Geometry(Graduate Texts in Mathematics, 168). Springer, New York, 1996.10.1007/978-1-4612-4044-0 · Zbl 0869.52001 [7] K. R.Goodearl. Notes on Real and Complex C^∗ -Algebras(Shiva Mathematics Series, 5). Birkhäuser, Boston, 1982. [8] A.Guilloux. A brief remark on orbits of SL(2, ℤ) in the Euclidean plane. Ergod. Th. & Dynam. Sys.30 (2010), 1101-1109.10.1017/S0143385709000315 · Zbl 1203.37010 [9] M.Laurent and A.Nogueira. Approximation to points in the plane by SL(2, ℤ)-orbits. J. Lond. Math. Soc. (2)85 (2012), 409-429.10.1112/jlms/jdr061 · Zbl 1268.11092 [10] D.Mundici. Interpretation of AF C^∗ -algebras in Łukasiewicz sentential calculus. J. Funct. Anal.65 (1986), 15-63.10.1016/0022-1236(86)90015-7 · Zbl 0597.46059 [11] D.Mundici. Farey stellar subdivisions, ultrasimplicial groups, and K_0 of AF C^∗ -algebras. Adv. Math.68 (1988), 23-39.10.1016/0001-8708(88)90006-0 · Zbl 0678.06008 [12] D.Mundici. The Haar theorem for lattice-ordered abelian groups with order-unit. Discrete Contin. Dyn. Syst.21 (2008), 537-549.10.3934/dcds.2008.21.537 · Zbl 1154.28007 [13] D.Mundici. Revisiting the Farey AF algebra. Milan J. Math.79 (2011), 643-656.10.1007/s00032-011-0166-3 · Zbl 1269.46041 [14] D.Mundici. Invariant measure under the affine group over ℤ. Combin. Probab. Comput.23 (2014), 248-268.10.1017/S096354831300062X · Zbl 1298.52016 [15] A.Nogueira. Orbit distribution on ℝ^2 under the natural action of SL(2, ℤ). Indag. Math. (N.S.)13 (2002), 103-124.10.1016/S0019-3577(02)90009-1 · Zbl 1016.37003 [16] A.Nogueira. Lattice orbit distribution on ℝ^2. Ergod. Th. & Dynam. Sys.30 (2010), 1201-1214; Erratum, ibid., p. 1215.10.1017/S0143385709000558 · Zbl 1204.37004 [17] E.Witten. SL(2, ℤ) action on three-dimensional conformal field theories with abelian symmetry. From Fields to Strings: Circumnavigating Theoretical Physics(Ian Kogan Memorial Collection, 3). Ed. M.Shifman. World Scientific, Singapore, 2005, pp. 1173-1200.10.1142/9789812775344_0028 · Zbl 1160.81457 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.