Small generators of cocompact arithmetic Fuchsian groups

Authors:
Michelle Chu and Han Li

Journal:
Proc. Amer. Math. Soc. **144** (2016), 5121-5127

MSC (2010):
Primary 20H10, 11F06

DOI:
https://doi.org/10.1090/proc/13177

Published electronically:
June 30, 2016

MathSciNet review:
3556258

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In the study of Fuchsian groups, it is a nontrivial problem to determine a set of generators. Using a dynamical approach we construct for any cocompact arithmetic Fuchsian group a fundamental region in $\mathbf {SL}_2(\mathbb {R})$ from which we determine a set of small generators.

- A. Page,
*Methodes explicites pour les groupes arithmétiques*. Ph.D dissertation, Université de Bordeaux (2014). - Valentin Blomer and Farrell Brumley,
*On the Ramanujan conjecture over number fields*, Ann. of Math. (2)**174**(2011), no. 1, 581–605. MR**2811610**, DOI 10.4007/annals.2011.174.1.18 - Armand Borel and Harish-Chandra,
*Arithmetic subgroups of algebraic groups*, Ann. of Math. (2)**75**(1962), 485–535. MR**147566**, DOI 10.2307/1970210 - Marc Burger and Viktor Schroeder,
*Volume, diameter and the first eigenvalue of locally symmetric spaces of rank one*, J. Differential Geom.**26**(1987), no. 2, 273–284. MR**906391** - Ted Chinburg and Matthew Stover,
*Small generators for $S$-unit groups of division algebras*, New York J. Math.**20**(2014), 1175–1202. MR**3291615** - H. Jacquet and R. P. Langlands,
*Automorphic forms on $\textrm {GL}(2)$*, Lecture Notes in Mathematics, Vol. 114, Springer-Verlag, Berlin-New York, 1970. MR**0401654** - Stefan Johansson,
*On fundamental domains of arithmetic Fuchsian groups*, Math. Comp.**69**(2000), no. 229, 339–349. MR**1665958**, DOI 10.1090/S0025-5718-99-01167-9 - Dubi Kelmer and Lior Silberman,
*A uniform spectral gap for congruence covers of a hyperbolic manifold*, Amer. J. Math.**135**(2013), no. 4, 1067–1085. MR**3086069**, DOI 10.1353/ajm.2013.0039 - Henry H. Kim,
*Functoriality for the exterior square of $\textrm {GL}_4$ and the symmetric fourth of $\textrm {GL}_2$*, J. Amer. Math. Soc.**16**(2003), no. 1, 139–183. With appendix 1 by Dinakar Ramakrishnan and appendix 2 by Kim and Peter Sarnak. MR**1937203**, DOI 10.1090/S0894-0347-02-00410-1 - D. Y. Kleinbock and G. A. Margulis,
*Bounded orbits of nonquasiunipotent flows on homogeneous spaces*, Sinaĭ’s Moscow Seminar on Dynamical Systems, Amer. Math. Soc. Transl. Ser. 2, vol. 171, Amer. Math. Soc., Providence, RI, 1996, pp. 141–172. MR**1359098**, DOI 10.1090/trans2/171/11 - D. H. Lehmer,
*Factorization of certain cyclotomic functions*, Ann. of Math. (2)**34**(1933), no. 3, 461–479. MR**1503118**, DOI 10.2307/1968172 - Melissa L. Macasieb,
*Derived arithmetic Fuchsian groups of genus two*, Experiment. Math.**17**(2008), no. 3, 347–369. MR**2455706** - Colin Maclachlan and Alan W. Reid,
*The arithmetic of hyperbolic 3-manifolds*, Graduate Texts in Mathematics, vol. 219, Springer-Verlag, New York, 2003. MR**1937957**, DOI 10.1007/978-1-4757-6720-9 - Carlos Matheus,
*Some quantitative versions of Ratner’s mixing estimates*, Bull. Braz. Math. Soc. (N.S.)**44**(2013), no. 3, 469–488. MR**3124746**, DOI 10.1007/s00574-013-0022-x - Walter D. Neumann and Alan W. Reid,
*Arithmetic of hyperbolic manifolds*, Topology ’90 (Columbus, OH, 1990) Ohio State Univ. Math. Res. Inst. Publ., vol. 1, de Gruyter, Berlin, 1992, pp. 273–310. MR**1184416** - Marina Ratner,
*The rate of mixing for geodesic and horocycle flows*, Ergodic Theory Dynam. Systems**7**(1987), no. 2, 267–288. MR**896798**, DOI 10.1017/S0143385700004004 - John Voight,
*Computing fundamental domains for Fuchsian groups*, J. Théor. Nombres Bordeaux**21**(2009), no. 2, 469–491 (English, with English and French summaries). MR**2541438** - Paul Voutier,
*An effective lower bound for the height of algebraic numbers*, Acta Arith.**74**(1996), no. 1, 81–95. MR**1367580**, DOI 10.4064/aa-74-1-81-95

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Additional Information

**Michelle Chu**

Affiliation:
Department of Mathematics, University of Texas at Austin, Austin, Texas 78750

MR Author ID:
974106

Email:
mchu@math.utexas.edu

**Han Li**

Affiliation:
Department of Mathematics and Computer Sciences, Wesleyan University, Middletown, Connecticut 06457

MR Author ID:
1080132

Email:
hli03@wesleyan.edu

Received by editor(s):
February 8, 2015

Received by editor(s) in revised form:
December 23, 2015, and February 23, 2016

Published electronically:
June 30, 2016

Additional Notes:
The first author was supported in part by NSF Grant DMS-1148490.

The second author was supported in part by an AMS Simons Travel Grant.

Communicated by:
Nimish Shah

Article copyright:
© Copyright 2016
American Mathematical Society