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# Forum Mathematicum

Managing Editor: Bruinier, Jan Hendrik

Ed. by Blomer, Valentin / Cohen, Frederick R. / Droste, Manfred / Duzaar, Frank / Echterhoff, Siegfried / Frahm, Jan / Gordina, Maria / Shahidi, Freydoon / Sogge, Christopher D. / Takayama, Shigeharu / Wienhard, Anna

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Volume 28, Issue 5

# The hyperbolic lattice point problem in conjugacy classes

Dimitrios Chatzakos
• Department of Mathematics, University College London, Gower Street, London WC1E 6BT, United Kingdom of Great Britain and Northern Ireland
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/ Yiannis N. Petridis
• Corresponding author
• Department of Mathematics, University College London, Gower Street, London WC1E 6BT, United Kingdom of Great Britain and Northern Ireland
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Published Online: 2016-01-10 | DOI: https://doi.org/10.1515/forum-2015-0102

## Abstract

For Γ a cocompact or cofinite Fuchsian group, we study the hyperbolic lattice point problem in conjugacy classes, which is a modification of the classical hyperbolic lattice point problem. We use large sieve inequalities for the Riemann surfaces $\mathrm{\Gamma }\ℍ$ to obtain average results for the error term, which are conjecturally optimal. We give a new proof of the error bound $O\left({X}^{2/3}\right)$, due to Good. For ${\mathrm{SL}}_{2}\left(ℤ\right)$ we interpret our results in terms of indefinite quadratic forms.

Keywords: Lattice points; hyperbolic space; geodesic arcs

MSC 2010: 11F72; 37C35; 37D40

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Revised: 2015-11-04

Published Online: 2016-01-10

Published in Print: 2016-09-01

The first author was supported by a DTA from EPSRC during his PhD studies at UCL.

Citation Information: Forum Mathematicum, Volume 28, Issue 5, Pages 981–1003, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741,

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