Counting problems for geodesics on arithmetic hyperbolic surfaces

It is a longstanding problem to determine the precise relationship between the geodesic length spectrum of a hyperbolic manifold and its commensurability class. A well known result of Reid, for instance, shows that the geodesic length spectrum of an arithmetic hyperbolic surface determines the surface's commensurability class. It is known, however, that non-commensurable arithmetic hyperbolic surfaces may share arbitrarily large portions of their length spectra. In this paper we investigate this phenomenon and prove a number of quantitative results about the maximum cardinality of a family of pairwise non-commensurable arithmetic hyperbolic surfaces whose length spectra all contain a fixed (finite) set of nonnegative real numbers

On fields of definition of arithmetic Kleinian reflection groups II

Following the previous work of Nikulin and Agol, Belolipetsky, Storm, and Whyte it is known that there exist only finitely many (totally real) number fields that can serve as fields of definition of arithmetic hyperbolic reflection groups. We prove a new bound on the degree $n_k$ of these fields in dimension 3: $n_k$ does not exceed 9. Combined with previous results of Maclachlan and Nikulin, this leads to a new bound $n_k \le 25$ which is valid for all dimensions. We also obtain upper bounds for the discriminants of these fields and give some heuristic results which may be useful for the classification of arithmetic hyperbolic reflection groups.Comment: 9 pages, final version, equation numbering changed, to appear in IMR

Counting isospectral manifolds

Given a simple Lie group $H$ of real rank at least $2$ we show that the maximum cardinality of a set of isospectral non-isometric $H$-locally symmetric spaces of volume at most $x$ grows at least as fast as $x^{c\log x/ (\log\log x)^2}$ where $c = c(H)$ is a positive constant. In contrast with the real rank $1$ case, this bound comes surprisingly close to the total number of such spaces as estimated in a previous work of Belolipetsky and Lubotzky [BL]. Our proof uses Sunada's method, results of [BL], and some deep results from number theory. We also discuss an open number-theoretical problem which would imply an even faster growth estimate.Comment: 9 pages; v2: one reference added, this is a final version; v3 includes small corrections to the published versio