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Counting non-uniform lattices

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Abstract

In [BGLM] and [GLNP] it was conjectured that if H is a simple Lie group of real rank at least 2, then the number of conjugacy classes of (arithmetic) lattices in H of covolume at most x is x(γ(H)+o(1)) log x/ log log x where γ(H) is an explicit constant computable from the (absolute) root system of H. In [BLu] we disproved this conjecture. In this paper we prove that for most groups H the conjecture is actually true if we restrict to counting only non-uniform lattices.

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Authors and Affiliations

  1. Instituto Nacional de Matemática Pura e Aplicada, Estrada Dona Castorina, 110, 22460-320, Rio de Janeiro, Brazil

    Mikhail Belolipetsky

  2. Institute of Mathematics, The Hebrew University of Jerusalem, Givat Ram, Jerusalem, 91904, Israel

    Alexander Lubotzky

Authors
  1. Mikhail Belolipetsky
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  2. Alexander Lubotzky
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Correspondence to Alexander Lubotzky.

Additional information

Dedicated to Aner Shalev on his 60th birthday

Belolipetsky is partially supported by the CNPq, the FAPERJ and the MATH-AmSud Program “18-Math-08”.

Luboztky was suppoted by the NSF, the ISF and the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant agreement No. 692854).

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Belolipetsky, M., Lubotzky, A. Counting non-uniform lattices. Isr. J. Math. 232, 201–229 (2019). https://doi.org/10.1007/s11856-019-1868-4

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  • DOI: https://doi.org/10.1007/s11856-019-1868-4

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