Citation for published item:wgeeD wihel nd yhD ree nd interD hle @PHIUA 9niform ongruene ounting for hottky semigroups in vP@AF9D tournl f¤ ur die reine und ngewndte wthemtikF a grelles journlF F Further information on publisher's website:The nal publication is available at www.degruyter.com With an appendix by Jean Bourgain, Alex Kontorovich and Michael Magee. Use policyThe full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. Abstract. Let Γ be a Schottky semigroup in SL2(Z), and for q ∈ N, let Γ(q) := {γ ∈ Γ : γ = e (mod q)} be its congruence subsemigroup of level q. Let δ denote the Hausdorff dimension of the limit set of Γ. We prove the following uniform congruence counting theorem with respect to the family of Euclidean norm balls BR in M2(R) of radius R: for all positive integer q with no small prime factors,as R → ∞ for some cΓ > 0, C > 0, > 0 which are independent of q. Our technique also applies to give a similar counting result for the continued fractions semigroup of SL2(Z), which arises in the study of Zaremba's conjecture on continued fractions.
We introduce a permutation model for random degree n covers Xn of a non-elementary convex-cocompact hyperbolic surface X = Γ\H. Let δ be the Hausdorff dimension of the limit set of Γ. We say that a resonance of Xn is new if it is not a resonance of X, and similarly define new eigenvalues of the Laplacian.We prove that for any > 0 and H > 0, with probability tending to 1 as n → ∞, there are no new resonances s = σ + it of Xn with σ ∈ [ 3 4 δ + , δ] and t ∈ [−H, H]. This implies in the case of δ > 1 2 that there is an explicit interval where there are no new eigenvalues of the Laplacian on Xn. By combining these results with a deterministic 'high frequency' resonance-free strip result, we obtain the corollary that there is an η = η(X) such that with probability → 1 as n → ∞, there are no new resonances of Xn in the region { s : Re(s) > δ − η }.
We establish an asymptotic formula for the number of integer solutions to the Markoff-
Let w be a word in the free group on r generators. The expected value of the trace of the word in r independent Haar elements of O(n) gives a function T r O w (n) of n. We show that T r O w (n) has a convergent Laurent expansion at n = ∞ involving maps on surfaces and L 2 -Euler characteristics of mapping class groups associated to these maps. This can be compared to known, by now classical, results for the GUE and GOE ensembles, and is similar to previous results concerning U (n), yet with some surprising twists.A priori to our result, T r O w (n) does not change if w is replaced with α(w) where α is an automorphism of the free group. One main feature of the Laurent expansion we obtain is that its coefficients respect this symmetry under Aut(F r ).As corollaries of our main theorem, we obtain a quantitative estimate on the rate of decay of T r O w (n) as n → ∞, we generalize a formula of Frobenius and Schur, and we obtain a universality result on random orthogonal matrices sampled according to words in free groups, generalizing a theorem of Diaconis and Shahshahani.Our results are obtained more generally for a tuple of words w 1 , . . . , w , leading to functions T r O w1,...,w . We also obtain all the analogous results for the compact symplectic groups Sp(n) through a rather mysterious duality formula.
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