Cheeger constants of hyperbolic reflection groups and Maass cusp forms of small eigenvalues

Benson, Brian; Lakeland, Grant; Then, Holger

doi:10.1090/proc/15152

Cheeger constants of hyperbolic reflection groups and Maass cusp forms of small eigenvalues
HTML articles powered by AMS MathViewer

by Brian A. Benson, Grant S. Lakeland and Holger Then

Proc. Amer. Math. Soc. 149 (2021), 417-438

DOI: https://doi.org/10.1090/proc/15152

Published electronically: October 9, 2020

PDF | Request permission

Abstract:

We compute the Cheeger constants of a collection of hyperbolic surfaces corresponding to maximal non-cocompact arithmetic Fuchsian groups, and to subgroups which are the rotation subgroup of maximal reflection groups. The Cheeger constants are geometric quantities, but relate to the smallest eigenvalues of Maass cusp forms. From geometrical considerations, we find evidence for the existence of small eigenvalues. We search for these small eigenvalues and compute the corresponding Maass cusp forms numerically.

References

Colin Adams and Frank Morgan, Isoperimetric curves on hyperbolic surfaces, Proc. Amer. Math. Soc. 127 (1999), no. 5, 1347–1356. MR 1487351, DOI 10.1090/S0002-9939-99-04778-4
Ian Agol, Finiteness of arithmetic Kleinian reflection groups, International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, 2006, pp. 951–960. MR 2275630
Ian Agol, Mikhail Belolipetsky, Peter Storm, and Kevin Whyte, Finiteness of arithmetic hyperbolic reflection groups, Groups Geom. Dyn. 2 (2008), no. 4, 481–498. MR 2442945, DOI 10.4171/GGD/47
Alan F. Beardon, The geometry of discrete groups, Graduate Texts in Mathematics, vol. 91, Springer-Verlag, New York, 1983. MR 698777, DOI 10.1007/978-1-4612-1146-4
Mikhail Belolipetsky, Finiteness theorems for congruence reflection groups, Transform. Groups 16 (2011), no. 4, 939–954. MR 2852486, DOI 10.1007/s00031-011-9156-3
Brian A. Benson, The Cheeger constant, isoperimetric problems, and hyperbolic surfaces, Preprint, arXiv:1509.08993, (2015).
Andrew R. Booker, Andreas Strömbergsson, and Holger Then, Bounds and algorithms for the $K$-Bessel function of imaginary order, LMS J. Comput. Math. 16 (2013), 78–108. MR 3044476, DOI 10.1112/S1461157013000028
Peter Buser, A note on the isoperimetric constant, Ann. Sci. École Norm. Sup. (4) 15 (1982), no. 2, 213–230. MR 683635, DOI 10.24033/asens.1426
Jeff Cheeger, A lower bound for the smallest eigenvalue of the Laplacian, Problems in analysis (Sympos. in honor of Salomon Bochner, Princeton Univ., Princeton, N.J., 1969) Princeton Univ. Press, Princeton, N.J., 1970, pp. 195–199. MR 0402831
E. Hecke, Über das Verhalten von und ähnlichen Funktionen bei Modulsubstitutionen, J. Reine Angew. Math. 157 (1927), 159–170 (German). MR 1581115, DOI 10.1515/crll.1927.157.159
Dennis A. Hejhal, On eigenfunctions of the Laplacian for Hecke triangle groups, Emerging applications of number theory (Minneapolis, MN, 1996) IMA Vol. Math. Appl., vol. 109, Springer, New York, 1999, pp. 291–315. MR 1691537, DOI 10.1007/978-1-4612-1544-8_{1}1
Dennis A. Hejhal and Andreas Strömbergsson, On quantum chaos and Maass waveforms of CM-type, Found. Phys. 31 (2001), no. 3, 519–533. Invited papers dedicated to Martin C. Gutzwiller, Part IV. MR 1839791, DOI 10.1023/A:1017521729782
Heinz Helling, Bestimmung der Kommensurabilitätsklasse der Hilbertschen Modulgruppe, Math. Z. 92 (1966), 269–280 (German). MR 228437, DOI 10.1007/BF01112194
Heinz Helling, On the commensurability class of the rational modular group, J. London Math. Soc. (2) 2 (1970), 67–72. MR 277620, DOI 10.1112/jlms/s2-2.1.67
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
Grant S. Lakeland, Arithmetic reflection groups and congruence subgroups, Ph.D. thesis, The University of Texas at Austin, 2012.
D. D. Long, C. Maclachlan, and A. W. Reid, Arithmetic Fuchsian groups of genus zero, Pure Appl. Math. Q. 2 (2006), no. 2, Special Issue: In honor of John H. Coates., 569–599. MR 2251482, DOI 10.4310/PAMQ.2006.v2.n2.a9
Hans Maass, Über eine neue Art von nichtanalytischen automorphen Funktionen und die Bestimmung Dirichletscher Reihen durch Funktionalgleichungen, Math. Ann. 121 (1949), 141–183 (German). MR 31519, DOI 10.1007/BF01329622
V. V. Nikulin, Finiteness of the number of arithmetic groups generated by reflections in Lobachevskiĭ spaces, Izv. Ross. Akad. Nauk Ser. Mat. 71 (2007), no. 1, 55–60 (Russian, with Russian summary); English transl., Izv. Math. 71 (2007), no. 1, 53–56. MR 2477273, DOI 10.1070/IM2007v071n01ABEH002349
Walter Roelcke, Das Eigenwertproblem der automorphen Formen in der hyperbolischen Ebene, I, Math. Ann. 167 (1966), no. 4, 292–337 (German). MR 1513277, DOI 10.1007/BF01364540
W. Roelcke, Das Eigenwertproblem der automorphen Formen in der hyperbolischen Ebene. II, Math. Annalen 168 (1967), 261–324.
Björn Selander and Andreas Strömbergsson, Sextic coverings of genus two which are branched at three points, 2002, preprint, http://www2.math.uu.se/~astrombe/papers/papers.html.
Atle Selberg, On the estimation of Fourier coefficients of modular forms, Proc. Sympos. Pure Math., Vol. VIII, Amer. Math. Soc., Providence, R.I., 1965, pp. 1–15. MR 0182610
Andreas Strömbergsson, A pullback algorithm for general (cofinite) Fuchsian groups, 2000, http://www2.math.uu.se/~astrombe/papers/pullback.ps.
Holger Then, Maass cusp forms for large eigenvalues, Math. Comp. 74 (2005), no. 249, 363–381. MR 2085897, DOI 10.1090/S0025-5718-04-01658-8
H. Then, Large sets of consecutive Maass forms and fluctuations in the Weyl remainder, preprint, arXiv:1212.3149, (2012).