On the number of lattice points in a ball in three-dimensional Lobachevskian space
Let \(\Delta= y^2 (\partial^2/ \partial x^2_1 +\partial^2/ \partial x^2_2+ \partial^2/ \partial y^2)- y\partial/ \partial y\) be the Laplace-Beltrami operator for the hyperbolic metric on the three-dimensional hyperbolic space \(\mathbb{H}^3 =\{(x_1,x_2,y) \in\mathbb{R}^3: y>0\}\). The aim of the paper under review is to sketch a proof of the Theorem: The smallest positive eigenvalue \(\lambda_1\) of \(-\Delta\) on \(L^2 (\text{PSL} (2,\mathbb{Z}[i]) \backslash \mathbb{H}^3)\) satisfies \[ \lambda_1> 1-(1-\rho)^2 \] for some \(\rho\geq 1/128\). The idea of proof is as follows: The authors estimate the number of lattice points for \(\Gamma= \text{PSL} (2,\mathbb{Z} [i])\) in a hyperbolic ball of radius \(R\) by means of the method of trigonometric sums. The known asymptotic behaviour of the number of lattice points as \(R\to\infty\) yields the estimate for \(\lambda_1\). (Reviewer's remark. Better results are known from the work of \textit{K. Stramm}: Kleine Eigenwerte des Laplace-Operators zu Kongruenzgruppen, Schriftenr. Math. Inst. Univ. Münster, 3. Ser., H. 11 (1994; Zbl 0842.11024). Stramm proves that \(\lambda_1>1\) for the groups \(\text{PSL} (2,{\mathfrak o}_d)\) where \({\mathfrak o}_d\) is the ring of integers of \(\mathbb{Q} (\sqrt{-d})\) and \(d\in \{1,2,3,7,11\}\). The authors' result even falls short of Sarnak's result \(\lambda_1 \geq {3\over 4}\), which holds for all congruence subgroups of all groups \(\text{PSL} (2,{\mathfrak o}_d)\) [see \textit{P. Sarnak}, Acta Math. 151, 253-295 (1983; Zbl 0527.10022)]. Item [5] from the authors' references is now available in print; see \textit{M. Suogui}, Mosc. Univ. Math. Bull. 48, No. 1, 6-11 (1993); translation from Vestn. Mosk. Univ., Ser. I 1993, No. 1, 8-14 (1993; Zbl 0846.11057)].
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