Splitting density for lifting about discrete groups. (English) Zbl 1148.11026
Let \(\pi:\widetilde X\to X\) be a finite unramified covering of hyperbolic Riemannian surfaces of finite volume. For a given closed geodesic \(c\) in \(X\) the pre-image \(\pi^{-1}(c)\) consists of finitely many closed geodesics of lengths \(e_1l(c),\dots,e_kl(c)\), where \(l(c)\) is the length of \(c\). The tuple \(e(c)=(e_1,\dots,e_k)\) of natural numbers is a partition of the degree \(n=[\widetilde X:X]\). For a given partition \(p\) of \(n\) the authors give an asymptotic formula for the function
\[ \#\{ c: l(c)\leq x,\;e(c)=p\}, \]
as \(x\to\infty\). If the covering \(\pi\) is normal, this asymptotic is known as a result of the trace formula. The current result is a quickly drawn consequence of the normal case.
The authors fill the rest of the paper with explicit computations of examples such as congruence groups. Reviewer’s remarks: The paper lacks precision as some notations are not explained and some results are stated in much larger generality than they can be true.
\[ \#\{ c: l(c)\leq x,\;e(c)=p\}, \]
as \(x\to\infty\). If the covering \(\pi\) is normal, this asymptotic is known as a result of the trace formula. The current result is a quickly drawn consequence of the normal case.
The authors fill the rest of the paper with explicit computations of examples such as congruence groups. Reviewer’s remarks: The paper lacks precision as some notations are not explained and some results are stated in much larger generality than they can be true.
Reviewer: Anton Deitmar (Tübingen)
MSC:
| 11F72 | Spectral theory; trace formulas (e.g., that of Selberg) |
Keywords:
prime geodesic theorem; splitting density; Selberg’s zeta function; regular cover; congruence subgroup; semisimple Lie groupsReferences:
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