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Crystalline flow starting from a general polygon
Equidistribution of translates of a homogeneous measure on the Borel–Serre compactification
Beijing International Center for Mathematical Research, Peking University, No. 5 Yiheyuan Road Haidian District, Beijing, 100871, China |
Let $ \mathit{\boldsymbol{\mathrm{G}}} $ be a semisimple linear algebraic group defined over rational numbers, $ \mathrm{K} $ be a maximal compact subgroup of its real points and $ \Gamma $ be an arithmetic lattice. One can associate a probability measure $ \mu_{ \mathrm{H}} $ on $ \Gamma \backslash \mathrm{G} $ for each subgroup $ \mathit{\boldsymbol{\mathrm{H}}} $ of $ \mathit{\boldsymbol{\mathrm{G}}} $ defined over $ \mathbb{Q} $ with no non-trivial rational characters. As G acts on $ \Gamma \backslash \mathrm{G} $ from the right, we can push forward this measure by elements from $ \mathrm{G} $. By pushing down these measures to $ \Gamma \backslash \mathrm{G}/ \mathrm{K} $, we call them homogeneous. It is a natural question to ask what are the possible weak-$ * $ limits of homogeneous measures. In the non-divergent case this has been answered by Eskin–Mozes–Shah. In the divergent case Daw–Gorodnik–Ullmo prove a refined version in some non-trivial compactifications of $ \Gamma \backslash \mathrm{G}/ \mathrm{K} $ for $ \mathit{\boldsymbol{\mathrm{H}}} $ generated by real unipotents. In the present article we build on their work and generalize the theorem to the case of general $ \mathit{\boldsymbol{\mathrm{H}}} $ with no non-trivial rational characters. Our results rely on (1) a non-divergent criterion on $ {\text{SL}}_n $ proved by geometry of numbers and a theorem of Kleinbock–Margulis; (2) relations between partial Borel–Serre compactifications associated with different groups proved by geometric invariant theory and reduction theory. 193 words.
References:
| [1] |
A. Borel, Introduction to Arithmetic Groups, University Lectures Series, vol. 73, American Mathematical Soc., 2019.
doi: 10.1090/ulect/073. |
| [2] |
A. Borel and L. Ji, Compactifications of Symmetric and Locally Symmetric Spaces, Mathematics: Theory & Applications, Birkhäuser Boston, Inc., Boston, MA, 2006. |
| [3] |
A. Borel and J.-P. Serre,
Corners and arithmetic groups, Comment. Math. Helv., 48 (1973), 436-491.
doi: 10.1007/BF02566134. |
| [4] |
S. G. Dani and G. A. Margulis,
Asymptotic behaviour of trajectories of unipotent flows on homogeneous spaces, Proc. Indian Acad. Sci. Math. Sci., 101 (1991), 1-17.
doi: 10.1007/BF02872005. |
| [5] |
S. G. Dani and G. A. Margulis,
Limit distributions of orbits of unipotent flows and values of quadratic forms, I. M. Gel$\overset{'}{\mathop{f}}$and Seminar, Adv. Soviet Math., Amer. Math. Soc., Providence, RI, 16 (1993), 91-137.
|
| [6] |
C. Daw, A. Gorodnik and E. Ullmo,
Convergence of measures on compactifications of locally symmetric spaces, Math. Z., 297 (2021), 1293-1328.
doi: 10.1007/s00209-020-02558-w. |
| [7] |
C. Daw, A. Gorodnik, E. Ullmo and J. Li, The Space of Homogeneous Probability Measures on $\overline{\Gamma\backslash X}^S_{ \rm max }$ is Compact, e-prints, 2019, arXiv: 1910.04568. |
| [8] |
W. Duke, Z. Rudnick and P. Sarnak,
Density of integer points on affine homogeneous varieties, Duke Math. J., 71 (1993), 143-179.
doi: 10.1215/S0012-7094-93-07107-4. |
| [9] |
A. Eskin and C. McMullen,
Mixing, counting, and equidistribution in Lie groups, Duke Math. J., 71 (1993), 181-209.
doi: 10.1215/S0012-7094-93-07108-6. |
| [10] |
A. Eskin, G. Margulis and S. Mozes,
Upper bounds and asymptotics in a quantitative version of the Oppenheim conjecture, Ann. of Math., 147 (1998), 93-141.
doi: 10.2307/120984. |
| [11] |
A. Eskin, S. Mozes and N. Shah,
Unipotent flows and counting lattice points on homogeneous varieties, Ann. of Math., 143 (1996), 253-299.
doi: 10.2307/2118644. |
| [12] |
A. Eskin, S. Mozes and N. Shah,
Non-divergence of translates of certain algebraic measures, Geom. Funct. Anal., 7 (1997), 48-80.
doi: 10.1007/PL00001616. |
| [13] |
D. Y. Kleinbock and G. A. Margulis,
Flows on homogeneous spaces and Diophantine approximation on manifolds, Ann. of Math., 148 (1998), 339-360.
doi: 10.2307/120997. |
| [14] |
B. Klingler and A. Yafaev,
The André-Oort conjecture, Ann. of Math., 180 (2014), 867-925.
doi: 10.4007/annals.2014.180.3.2. |
| [15] |
G. D. Mostow,
Self-adjoint groups, Ann. of Math., 62 (1955), 44-55.
doi: 10.2307/2007099. |
| [16] |
S. Mozes and N. Shah,
On the space of ergodic invariant measures of unipotent flows, Ergodic Theory Dynam. Systems, 15 (1995), 149-159.
doi: 10.1017/S0143385700008282. |
| [17] |
D. Mumford, J. Fogarty and F. Kirwan, Geometric Invariant Theory, 3$^{rd}$ edition, Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)], vol. 34, Springer-Verlag, Berlin, 1994. |
| [18] |
M. Ratner,
On Raghunathan's measure conjecture, Ann. of Math., 134 (1991), 545-607.
doi: 10.2307/2944357. |
| [19] |
T. A. Springer, Linear Algebraic Groups, 2$^{nd}$ edition, Progress in Mathematics, vol. 9, Birkhäuser Boston, Inc., Boston, MA, 1998.
doi: 10.1007/978-0-8176-4840-4. |
| [20] |
E. Ullmo and A. Yafaev,
Galois orbits and equidistribution of special subvarieties: Towards the André-Oort conjecture, Ann. of Math., 180 (2014), 823-865.
doi: 10.4007/annals.2014.180.3.1. |
| [21] |
R. Zhang, Translates of homogeneous measures associated with Observable Subgroups on some homogeneous spaces, arXiv e-prints, 2020, arXiv: 1909.02666. |
show all references
References:
| [1] |
A. Borel, Introduction to Arithmetic Groups, University Lectures Series, vol. 73, American Mathematical Soc., 2019.
doi: 10.1090/ulect/073. |
| [2] |
A. Borel and L. Ji, Compactifications of Symmetric and Locally Symmetric Spaces, Mathematics: Theory & Applications, Birkhäuser Boston, Inc., Boston, MA, 2006. |
| [3] |
A. Borel and J.-P. Serre,
Corners and arithmetic groups, Comment. Math. Helv., 48 (1973), 436-491.
doi: 10.1007/BF02566134. |
| [4] |
S. G. Dani and G. A. Margulis,
Asymptotic behaviour of trajectories of unipotent flows on homogeneous spaces, Proc. Indian Acad. Sci. Math. Sci., 101 (1991), 1-17.
doi: 10.1007/BF02872005. |
| [5] |
S. G. Dani and G. A. Margulis,
Limit distributions of orbits of unipotent flows and values of quadratic forms, I. M. Gel$\overset{'}{\mathop{f}}$and Seminar, Adv. Soviet Math., Amer. Math. Soc., Providence, RI, 16 (1993), 91-137.
|
| [6] |
C. Daw, A. Gorodnik and E. Ullmo,
Convergence of measures on compactifications of locally symmetric spaces, Math. Z., 297 (2021), 1293-1328.
doi: 10.1007/s00209-020-02558-w. |
| [7] |
C. Daw, A. Gorodnik, E. Ullmo and J. Li, The Space of Homogeneous Probability Measures on $\overline{\Gamma\backslash X}^S_{ \rm max }$ is Compact, e-prints, 2019, arXiv: 1910.04568. |
| [8] |
W. Duke, Z. Rudnick and P. Sarnak,
Density of integer points on affine homogeneous varieties, Duke Math. J., 71 (1993), 143-179.
doi: 10.1215/S0012-7094-93-07107-4. |
| [9] |
A. Eskin and C. McMullen,
Mixing, counting, and equidistribution in Lie groups, Duke Math. J., 71 (1993), 181-209.
doi: 10.1215/S0012-7094-93-07108-6. |
| [10] |
A. Eskin, G. Margulis and S. Mozes,
Upper bounds and asymptotics in a quantitative version of the Oppenheim conjecture, Ann. of Math., 147 (1998), 93-141.
doi: 10.2307/120984. |
| [11] |
A. Eskin, S. Mozes and N. Shah,
Unipotent flows and counting lattice points on homogeneous varieties, Ann. of Math., 143 (1996), 253-299.
doi: 10.2307/2118644. |
| [12] |
A. Eskin, S. Mozes and N. Shah,
Non-divergence of translates of certain algebraic measures, Geom. Funct. Anal., 7 (1997), 48-80.
doi: 10.1007/PL00001616. |
| [13] |
D. Y. Kleinbock and G. A. Margulis,
Flows on homogeneous spaces and Diophantine approximation on manifolds, Ann. of Math., 148 (1998), 339-360.
doi: 10.2307/120997. |
| [14] |
B. Klingler and A. Yafaev,
The André-Oort conjecture, Ann. of Math., 180 (2014), 867-925.
doi: 10.4007/annals.2014.180.3.2. |
| [15] |
G. D. Mostow,
Self-adjoint groups, Ann. of Math., 62 (1955), 44-55.
doi: 10.2307/2007099. |
| [16] |
S. Mozes and N. Shah,
On the space of ergodic invariant measures of unipotent flows, Ergodic Theory Dynam. Systems, 15 (1995), 149-159.
doi: 10.1017/S0143385700008282. |
| [17] |
D. Mumford, J. Fogarty and F. Kirwan, Geometric Invariant Theory, 3$^{rd}$ edition, Ergebnisse der Mathematik und ihrer Grenzgebiete (2) [Results in Mathematics and Related Areas (2)], vol. 34, Springer-Verlag, Berlin, 1994. |
| [18] |
M. Ratner,
On Raghunathan's measure conjecture, Ann. of Math., 134 (1991), 545-607.
doi: 10.2307/2944357. |
| [19] |
T. A. Springer, Linear Algebraic Groups, 2$^{nd}$ edition, Progress in Mathematics, vol. 9, Birkhäuser Boston, Inc., Boston, MA, 1998.
doi: 10.1007/978-0-8176-4840-4. |
| [20] |
E. Ullmo and A. Yafaev,
Galois orbits and equidistribution of special subvarieties: Towards the André-Oort conjecture, Ann. of Math., 180 (2014), 823-865.
doi: 10.4007/annals.2014.180.3.1. |
| [21] |
R. Zhang, Translates of homogeneous measures associated with Observable Subgroups on some homogeneous spaces, arXiv e-prints, 2020, arXiv: 1909.02666. |
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