Geometry & Topology Volume 27, issue 2 (2023)

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Anosov groups: local mixing, counting and equidistribution

Samuel Edwards, Minju Lee and Hee Oh

Geometry & Topology 27 (2023) 513–573

DOI: 10.2140/gt.2023.27.513

Bibliography

1	Y Benoist, Propriétés asymptotiques des groupes linéaires, Geom. Funct. Anal. 7 (1997) 1 MR1437472
2	Y Benoist, H Oh, Effective equidistribution of S–integral points on symmetric varieties, Ann. Inst. Fourier (Grenoble) 62 (2012) 1889 MR3025156
3	J Bochi, R Potrie, A Sambarino, Anosov representations and dominated splittings, J. Eur. Math. Soc. 21 (2019) 3343 MR4012341
4	M Bridgeman, R Canary, F Labourie, A Sambarino, The pressure metric for Anosov representations, Geom. Funct. Anal. 25 (2015) 1089 MR3385630
5	M Burger, Horocycle flow on geometrically finite surfaces, Duke Math. J. 61 (1990) 779 MR1084459
6	L Carvajales, Counting problems for special-orthogonal Anosov representations, Ann. Inst. Fourier (Grenoble) 70 (2020) 1199 MR4117059
7	L Carvajales, Growth of quadratic forms under Anosov subgroups, Int. Math. Res. Not. 2023 (2023) 785 MR4530121
8	M Chow, P Sarkar, Local mixing of one-parameter diagonal flows on Anosov homogeneous spaces, Int. Math. Res. Not. (2023)
9	N T Dang, O Glorieux, Topological mixing of Weyl chamber flows, Ergodic Theory Dynam. Systems 41 (2021) 1342 MR4240601
10	W Duke, Z Rudnick, P Sarnak, Density of integer points on affine homogeneous varieties, Duke Math. J. 71 (1993) 143 MR1230289
11	A Eskin, C McMullen, Mixing, counting, and equidistribution in Lie groups, Duke Math. J. 71 (1993) 181 MR1230290
12	V Fock, A Goncharov, Moduli spaces of local systems and higher Teichmüller theory, Publ. Math. Inst. Hautes Études Sci. 103 (2006) 1 MR2233852
13	A Gorodnik, H Oh, Orbits of discrete subgroups on a symmetric space and the Furstenberg boundary, Duke Math. J. 139 (2007) 483 MR2350851
14	A Gorodnik, H Oh, N Shah, Integral points on symmetric varieties and Satake compactifications, Amer. J. Math. 131 (2009) 1 MR2488484
15	A Gorodnik, H Oh, N Shah, Strong wavefront lemma and counting lattice points in sectors, Israel J. Math. 176 (2010) 419 MR2653201
16	F Guéritaud, O Guichard, F Kassel, A Wienhard, Anosov representations and proper actions, Geom. Topol. 21 (2017) 485 MR3608719
17	O Guichard, A Wienhard, Anosov representations: domains of discontinuity and applications, Invent. Math. 190 (2012) 357 MR2981818
18	Y Guivarc’h, A Raugi, Actions of large semigroups and random walks on isometric extensions of boundaries, Ann. Sci. École Norm. Sup. 40 (2007) 209 MR2339285
19	N J Hitchin, Lie groups and Teichmüller space, Topology 31 (1992) 449 MR1174252
20	R E Howe, C C Moore, Asymptotic properties of unitary representations, J. Functional Analysis 32 (1979) 72 MR533220
21	M Kapovich, B Leeb, J Porti, Anosov subgroups: dynamical and geometric characterizations, Eur. J. Math. 3 (2017) 808 MR3736790
22	F Kassel, Geometric structures and representations of discrete groups, from: "Proceedings of the International Congress of Mathematicians" (editors B Sirakov, P N de Souza, M Viana), World Sci. (2018) 1133 MR3966802
23	A W Knapp, Lie groups beyond an introduction, 140, Birkhäuser (2002) MR1920389
24	F Labourie, Anosov flows, surface groups and curves in projective space, Invent. Math. 165 (2006) 51 MR2221137
25	M Lee, H Oh, Invariant measures for horospherical actions and Anosov groups, Int. Math. Res. Not. (2022)
26	M Lee, H Oh, Ergodic decompositions of geometric measures on Anosov homogeneous spaces, preprint (2020) arXiv:2010.11337
27	G A Margulis, On some aspects of the theory of Anosov systems, Springer (2004) MR2035655
28	T Matsuki, The orbits of affine symmetric spaces under the action of minimal parabolic subgroups, J. Math. Soc. Japan 31 (1979) 331 MR527548
29	A Mohammadi, H Oh, Matrix coefficients, counting and primes for orbits of geometrically finite groups, J. Eur. Math. Soc. 17 (2015) 837 MR3336838
30	H Oh, Uniform pointwise bounds for matrix coefficients of unitary representations and applications to Kazhdan constants, Duke Math. J. 113 (2002) 133 MR1905394
31	H Oh, W Pan, Local mixing and invariant measures for horospherical subgroups on abelian covers, Int. Math. Res. Not. 2019 (2019) 6036 MR4016891
32	H Oh, N A Shah, Equidistribution and counting for orbits of geometrically finite hyperbolic groups, J. Amer. Math. Soc. 26 (2013) 511 MR3011420
33	H Oh, D Winter, Uniform exponential mixing and resonance free regions for convex cocompact congruence subgroups of SL₂(ℤ), J. Amer. Math. Soc. 29 (2016) 1069 MR3522610
34	S J Patterson, The limit set of a Fuchsian group, Acta Math. 136 (1976) 241 MR450547
35	R Potrie, A Sambarino, Eigenvalues and entropy of a Hitchin representation, Invent. Math. 209 (2017) 885 MR3681396
36	J F Quint, Divergence exponentielle des sous-groupes discrets en rang supérieur, Comment. Math. Helv. 77 (2002) 563 MR1933790
37	J F Quint, Mesures de Patterson–Sullivan en rang supérieur, Geom. Funct. Anal. 12 (2002) 776 MR1935549
38	J F Quint, L’indicateur de croissance des groupes de Schottky, Ergodic Theory Dynam. Systems 23 (2003) 249 MR1971205
39	J F Quint, Propriété de Kazhdan et sous-groupes discrets de covolume infini, from: "Proceedings of the conference on harmonic analysis", Trav. Math. 14, Univ. Luxembourg (2003) 143 MR2065799
40	J F Quint, Groupes de Schottky et comptage, Ann. Inst. Fourier (Grenoble) 55 (2005) 373 MR2147895
41	J F Quint, An overview of Patterson–Sullivan theory, lecture notes (2006)
42	T Roblin, Ergodicité et équidistribution en courbure négative, 95, Soc. Math. France (2003) MR2057305
43	A Sambarino, Hyperconvex representations and exponential growth, Ergodic Theory Dynam. Systems 34 (2014) 986 MR3199802
44	A Sambarino, Quantitative properties of convex representations, Comment. Math. Helv. 89 (2014) 443 MR3229035
45	A Sambarino, The orbital counting problem for hyperconvex representations, Ann. Inst. Fourier (Grenoble) 65 (2015) 1755 MR3449196
46	H Schlichtkrull, Hyperfunctions and harmonic analysis on symmetric spaces, 49, Birkhäuser (1984) MR757178
47	D Sullivan, The density at infinity of a discrete group of hyperbolic motions, Inst. Hautes Études Sci. Publ. Math. 50 (1979) 171 MR556586
48	X Thirion, Propriétés de mélange du flot des chambres de Weyl des groupes de ping-pong, Bull. Soc. Math. France 137 (2009) 387 MR2574089
49	J Tits, Free subgroups in linear groups, J. Algebra 20 (1972) 250 MR286898
50	A Wienhard, An invitation to higher Teichmüller theory, from: "Proceedings of the International Congress of Mathematicians" (editors B Sirakov, P N de Souza, M Viana), World Sci. (2018) 1031 MR3966798
51	D Winter, Mixing of frame flow for rank one locally symmetric spaces and measure classification, Israel J. Math. 210 (2015) 467 MR3430281