On the uniform equidistribution of closed horospheres in hyperbolic manifolds (Q2909054)

The author proves asymptotic equidistribution results for pieces of large closed horospheres in cofinite hyperbolic manifolds of finite volume in arbitrary dimension. This extends earlier results by \textit{D. A. Hejhal} [Asian J. Math. 4, No. 4, 839--854 (2000; Zbl 1014.11038)] and \textit{A. Strömbergsson} [Duke Math. J. 123, No. 3, 507--547 (2004; Zbl 1060.37023)] in dimension 2 to manifolds of arbitrary dimension.NEWLINENEWLINE Hejhal proved that given a subsegment of length \(\ell_1<\ell\) of a closed horocycle of length \(\ell\) in a non-compact hyperbolic surface of finite volume, if \(\ell_1\geq\ell^{c+ \varepsilon}\), where \(\varepsilon> 0\) and \(c\geq{2\over 3}\) is a constant that only depends on \(M\), then the subsegment becomes equidistributed on \(M\). This result was improved by Strömbergsson, who proved that \(c\) can be set to \({1\over 2}\) independently of \(M\) and that this constant is optimal.NEWLINENEWLINE The author extends this result to manifolds of arbitrary dimension, with the constant \({1\over 2}\).NEWLINENEWLINE The proofs of the author use spectral methods and lead to precise estimates on the rate of convergence to equidistribution.