This is a small survey paper about connections between the arithmetic and geometric properties in the case of arithmetic Fuchsian groups.
Ceci est un petit papier de synthèse sur les connections entre les propriétés arithmétiques et géométriques dans le cas de groupes fuchsiens arithmétiques.
@article{AFST_2014_6_23_5_1093_0, author = {Slavyana Geninska}, title = {On arithmetic {Fuchsian} groups and their characterizations}, journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques}, pages = {1093--1102}, publisher = {Universit\'e Paul Sabatier, Institut de Math\'ematiques}, address = {Toulouse}, volume = {Ser. 6, 23}, number = {5}, year = {2014}, doi = {10.5802/afst.1437}, mrnumber = {3294603}, language = {en}, url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1437/} }
TY - JOUR AU - Slavyana Geninska TI - On arithmetic Fuchsian groups and their characterizations JO - Annales de la Faculté des sciences de Toulouse : Mathématiques PY - 2014 SP - 1093 EP - 1102 VL - 23 IS - 5 PB - Université Paul Sabatier, Institut de Mathématiques PP - Toulouse UR - https://afst.centre-mersenne.org/articles/10.5802/afst.1437/ DO - 10.5802/afst.1437 LA - en ID - AFST_2014_6_23_5_1093_0 ER -
%0 Journal Article %A Slavyana Geninska %T On arithmetic Fuchsian groups and their characterizations %J Annales de la Faculté des sciences de Toulouse : Mathématiques %D 2014 %P 1093-1102 %V 23 %N 5 %I Université Paul Sabatier, Institut de Mathématiques %C Toulouse %U https://afst.centre-mersenne.org/articles/10.5802/afst.1437/ %R 10.5802/afst.1437 %G en %F AFST_2014_6_23_5_1093_0
Slavyana Geninska. On arithmetic Fuchsian groups and their characterizations. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 23 (2014) no. 5, pp. 1093-1102. doi : 10.5802/afst.1437. https://afst.centre-mersenne.org/articles/10.5802/afst.1437/
[1] Beardon (A.).— The Geometry of discrete groups, Graduate Texts in Mathematics 91, Springer-Verlag, New York (1995). | MR | Zbl
[2] Benoist (Y.).— Propriétés asymptotiques des groupes linéaires, Geom. Funct. Anal. 7, p. 1-47 (1997). | MR | Zbl
[3] Cohen (P.) and Wolfart (J.).— Modular embeddings for some non-arithmetic Fuchsian groups, Acta Arith. 61, no. 1, p. 93-110 (1990). | MR | Zbl
[4] Corlette (K.).— Archimedean superrigidity and hyperbolic geometry, Ann. of Math. (2) 135, p. 165-182 (1992). | MR | Zbl
[5] Dal’Bo (F.) and Kim (I.).— A criterion of conjugacy for Zariski dense subgroups, C. R. Acad. Sci. Paris, t. 330 série I, p. 647-650 (2000). | MR | Zbl
[6] Deligne (P.) and Mostow (G. D.).— Monodromy of hypergeometric functions and nonlattice integral monodromy, Publ. Math. IHES 63, p. 5-89 (1986). | Numdam | MR | Zbl
[7] Geninska (S.) and Leuzinger (E.).— A geometric characterization of arithmetic Fuchsian groups, Duke Math. J. 142, p. 111-125 (2008). | MR | Zbl
[8] Geninska (S.).— The limit set of subgroups of arithmetic groups in , arXiv:1001.1720, to appear in Groups Geom. Dyn..
[9] Geninska (S.).— Examples of infinite covolume subgroups of with big limit sets, Math. Zeit. 272, p. 389-404 (2012). | MR | Zbl
[10] Gromov (M.) and Piatetski-Shapiro (I.).— Nonarithmetic groups in Lobachevsky spaces, Publ. Math. IHES 66, p. 93-103 (1988). | Numdam | MR | Zbl
[11] Gromov (M.) and Schoen (R.).— Harmonic maps into singular spaces and p-adic superrigidity for lattices in groups of rank one, Publ. Math. IHES 76, p. 165-246 (1992). | Numdam | MR | Zbl
[12] Kapovich (M.).— Arithmetic aspects of self-similar groups, Groups Geom. Dyn. 6, no. 4, p. 737-754 (2012). | MR | Zbl
[13] Katok (S.).— Fuchsian Groups, Chicago Lectures in Math., University of Chicago Press, Chicago (1992). | MR | Zbl
[14] Link (G.).— Geometry and Dynamics of Discrete Isometry Groups of Higher Rank Symmetric Spaces, Geometriae Dedicata 122, no 1, p. 51-75 (2006). | MR | Zbl
[15] Luo (W.) and Sarnak (P.).— Number variance for arithmetic hyperbolic surfaces, Comm. Math. Phys. 161, p. 419-432 (1994). | MR | Zbl
[16] Maclachlan (C.) and Reid (A.).— The Arithmetic of Hyperbolic 3-Manifolds, Graduate Texts in Mathematics 219, Springer-Verlag (2003). | MR | Zbl
[17] Margulis (G. A.).— Discrete subgroups of semisimple Lie groups, Springer-Verlag, Berlin (1991). | MR | Zbl
[18] Witte Morris (D.).— Introduction to Arithmetic Groups, preprint, http://arxiv.org/abs/math/0106063.
[19] Mostow (G. D.).— On a remarkable class of polyhedra in complex hyperbolic space, Pacific J. Math. 86, p. 171-276 (1980). | MR | Zbl
[20] Sarnak (P.).— Arithmetic quantum chaos, Israel Math. Conf. Proc. 8, p. 183-236 (1995). | MR | Zbl
[21] Schmutz (P.).— Arithmetic groups and the length spectrum of Riemann surfaces, Duke Math. J. 84, p. 199-215 (1996). | MR | Zbl
[22] Schmutz Schaller (P.).— Geometry of Riemann surfaces based on closed geodesics, Bull. Amer. Math. Soc. 35, p. 193-214 (1998). | MR | Zbl
[23] Schmutz Schaller (P.) and J. Wolfart.— Semi-arithmetic Fuchsian groups and modular embeddings, J. London Math. Soc. (2) 61, p. 13-24 (2000). | MR | Zbl
[24] Takeuchi (K.).— A characterization of arithmetic Fuchsian groups, J. Math. Soc. Japan 27, p. 600-612 (1975). | MR | Zbl
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