# American Institute of Mathematical Sciences

September  2016, 36(9): 4723-4738. doi: 10.3934/dcds.2016005

## Classifying GL$(n,\mathbb{Z})$-orbits of points and rational subspaces

 1 Institute of Computer Languages Theory and Logic Group, Technische Universität Wien, Favoritenstrasse 9-11, A-1040 Vienna, Austria 2 Department of Mathematics and Computer Science “Ulisse Dini", University of Florence, Viale Morgagni 67/a, I-50134 Florence, Italy

Received  July 2015 Revised  March 2016 Published  May 2016

We first show that the subgroup of the abelian real group $\mathbb{R}$ generated by the coordinates of a point in $x\in\mathbb{R}^n$ completely classifies the GL$(n,\mathbb{Z})$-orbit of $x$. This yields a short proof of J.S. Dani's theorem: the GL$(n,\mathbb{Z})$-orbit of $x\in\mathbb{R}^n$ is dense iff $x_i/x_j\in \mathbb{R}\setminus \mathbb{Q}$ for some $i,j=1,\dots,n$. We then classify GL$(n,\mathbb{Z})$-orbits of rational affine subspaces $F$ of $\mathbb{R}^n$. We prove that the dimension of $F$ together with the volume of a special parallelotope associated to $F$ yields a complete classifier of the GL$(n,\mathbb{Z})$-orbit of $F$.
Citation: Leonardo Manuel Cabrer, Daniele Mundici. Classifying GL$(n,\mathbb{Z})$-orbits of points and rational subspaces. Discrete & Continuous Dynamical Systems - A, 2016, 36 (9) : 4723-4738. doi: 10.3934/dcds.2016005
##### References:
 [1] L. M. Cabrer and D. Mundici, Classifying orbits of the affine group over the integers,, Ergodic Theory Dynam. Systems, (2015).  doi: 10.1017/etds.2015.45.  Google Scholar [2] J. S. Dani, Density properties of orbits under discrete groups,, J. Indian Math. Soc., 39 (1975), 189.   Google Scholar [3] S. G. Dani and A. Nogueira, On $SL(n,\mathbbZ)_+$-orbits on $\mathbbR^n$ and positive integral solutions of linear inequalities,, J. of Number Theory, 129 (2009), 2526.  doi: 10.1016/j.jnt.2008.12.010.  Google Scholar [4] L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions,, CRC Press, (1992).   Google Scholar [5] G. Ewald, Combinatorial Convexity and Algebraic Geometry,, Grad. Texts in Math., (1996).  doi: 10.1007/978-1-4612-4044-0.  Google Scholar [6] H. Federer, Geometric Measure Theory,, Springer, (1969).   Google Scholar [7] A. Guilloux, A brief remark on orbits of $\mathsf{SL}(2,\mathbbZ)$ in the Euclidean plane},, Ergodic Theory Dynam. Systems, 30 (2010), 1101.  doi: 10.1017/S0143385709000315.  Google Scholar [8] G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers,, Fifth edition, (1979).   Google Scholar [9] M. Laurent and A. Nogueira, Approximation to points in the plane by $\mathsf{SL}(2, \mathbbZ)$-orbits,, J. London Math. Soc., 85 (2012), 409.  doi: 10.1112/jlms/jdr061.  Google Scholar [10] R. Morelli, The birational geometry of toric varieties,, J. Algebraic Geom., 5 (1996), 751.   Google Scholar [11] D. Mundici, The Haar theorem for lattice-ordered abelian groups with order-unit,, Discrete Contin. Dyn. Syst., 21 (2008), 537.  doi: 10.3934/dcds.2008.21.537.  Google Scholar [12] D. Mundici, Invariant measure under the affine group over $\mathbbZ$,, Combin. Probab. Comput., 23 (2014), 248.  doi: 10.1017/S096354831300062X.  Google Scholar [13] A. Nogueira, Orbit distribution on $\mathbbR^2$ under the natural action of $SL(2,\mathbbZ)$,, Indag. Math. (N.S.), 13 (2002), 103.  doi: 10.1016/S0019-3577(02)90009-1.  Google Scholar [14] A. Nogueira, Lattice orbit distribution on $\mathbbR^2$,, Ergodic Theory Dynam. Systems, 30 (2010), 1201.  doi: 10.1017/S0143385709000558.  Google Scholar [15] T. Oda, Convex Bodies and Algebraic Geometry. An Introduction to the Theory of Toric Varieties,, A Series of Modern Surveys in Mathematics, (1988).   Google Scholar [16] J. R. Stallings, Lectures on Polyhedral Topology,, Tata Inst. Fund. Res., (1967).   Google Scholar [17] E. Witten, $\mathsf{SL}(2, \mathbbZ)$ action on three-dimensional conformal field theories with abelian symmetry,, In: From Fields to Strings: Circumnavigating Theoretical Physics, 2 (2005), 1173.   Google Scholar [18] J. Włodarczyk, Decompositions of birational toric maps in blow-ups and blow-downs,, Trans. Amer. Math. Soc., 349 (1997), 373.  doi: 10.1090/S0002-9947-97-01701-7.  Google Scholar

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##### References:
 [1] L. M. Cabrer and D. Mundici, Classifying orbits of the affine group over the integers,, Ergodic Theory Dynam. Systems, (2015).  doi: 10.1017/etds.2015.45.  Google Scholar [2] J. S. Dani, Density properties of orbits under discrete groups,, J. Indian Math. Soc., 39 (1975), 189.   Google Scholar [3] S. G. Dani and A. Nogueira, On $SL(n,\mathbbZ)_+$-orbits on $\mathbbR^n$ and positive integral solutions of linear inequalities,, J. of Number Theory, 129 (2009), 2526.  doi: 10.1016/j.jnt.2008.12.010.  Google Scholar [4] L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions,, CRC Press, (1992).   Google Scholar [5] G. Ewald, Combinatorial Convexity and Algebraic Geometry,, Grad. Texts in Math., (1996).  doi: 10.1007/978-1-4612-4044-0.  Google Scholar [6] H. Federer, Geometric Measure Theory,, Springer, (1969).   Google Scholar [7] A. Guilloux, A brief remark on orbits of $\mathsf{SL}(2,\mathbbZ)$ in the Euclidean plane},, Ergodic Theory Dynam. Systems, 30 (2010), 1101.  doi: 10.1017/S0143385709000315.  Google Scholar [8] G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers,, Fifth edition, (1979).   Google Scholar [9] M. Laurent and A. Nogueira, Approximation to points in the plane by $\mathsf{SL}(2, \mathbbZ)$-orbits,, J. London Math. Soc., 85 (2012), 409.  doi: 10.1112/jlms/jdr061.  Google Scholar [10] R. Morelli, The birational geometry of toric varieties,, J. Algebraic Geom., 5 (1996), 751.   Google Scholar [11] D. Mundici, The Haar theorem for lattice-ordered abelian groups with order-unit,, Discrete Contin. Dyn. Syst., 21 (2008), 537.  doi: 10.3934/dcds.2008.21.537.  Google Scholar [12] D. Mundici, Invariant measure under the affine group over $\mathbbZ$,, Combin. Probab. Comput., 23 (2014), 248.  doi: 10.1017/S096354831300062X.  Google Scholar [13] A. Nogueira, Orbit distribution on $\mathbbR^2$ under the natural action of $SL(2,\mathbbZ)$,, Indag. Math. (N.S.), 13 (2002), 103.  doi: 10.1016/S0019-3577(02)90009-1.  Google Scholar [14] A. Nogueira, Lattice orbit distribution on $\mathbbR^2$,, Ergodic Theory Dynam. Systems, 30 (2010), 1201.  doi: 10.1017/S0143385709000558.  Google Scholar [15] T. Oda, Convex Bodies and Algebraic Geometry. An Introduction to the Theory of Toric Varieties,, A Series of Modern Surveys in Mathematics, (1988).   Google Scholar [16] J. R. Stallings, Lectures on Polyhedral Topology,, Tata Inst. Fund. Res., (1967).   Google Scholar [17] E. Witten, $\mathsf{SL}(2, \mathbbZ)$ action on three-dimensional conformal field theories with abelian symmetry,, In: From Fields to Strings: Circumnavigating Theoretical Physics, 2 (2005), 1173.   Google Scholar [18] J. Włodarczyk, Decompositions of birational toric maps in blow-ups and blow-downs,, Trans. Amer. Math. Soc., 349 (1997), 373.  doi: 10.1090/S0002-9947-97-01701-7.  Google Scholar
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