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Groups, Geometry, and Dynamics


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Volume 15, Issue 1, 2021, pp. 57–81
DOI: 10.4171/GGD/591

Published online: 2020-12-24

A higher moment formula for the Siegel–Veech transform over quotients by Hecke triangle groups

Samantha Fairchild[1]

(1) University of Washington, Seattle, USA

We compute higher moments of the Siegel–Veech transform over quotients of SL$(2,\mathbb{R})$ by the Hecke triangle groups. After fixing a normalization of the Haar measure on SL$(2,\mathbb{R})$ we use geometric results and linear algebra to create explicit integration formulas which give information about densities of $k$-tuples of vectors in discrete subsets of $\mathbb{R}^2$ which arise as orbits of Hecke triangle groups. This generalizes work of W. Schmidt on the variance of the Siegel transform over SL$(2,\mathbb{R})/$SL$(2,\mathbb{Z})$.

Keywords: Hecke triangle group, Siegel–Veech transform

Fairchild Samantha: A higher moment formula for the Siegel–Veech transform over quotients by Hecke triangle groups. Groups Geom. Dyn. 15 (2021), 57-81. doi: 10.4171/GGD/591