Systole of congruence coverings of arithmetic hyperbolic manifolds

  • Plinio G.P. Murillo

    Korea Institute for Advanced Study (KIAS), Seoul, Republic of Korea
Systole of congruence coverings of arithmetic hyperbolic manifolds cover
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Abstract

In this paper we prove that, for any arithmetic hyperbolic nn-manifold MM of the first type, the systole of most of the principal congruence coverings MIM_{I} satisfy

sys(MI)8n(n+1)log(vol(MI))c,\mathrm{sys}(M_{I})\geq \frac{8}{n(n+1)}\mathrm{log}(\mathrm{vol}(M_{I}))-c,

where cc is a constant independent of II. This generalizes previous work of Buser and Sarnak, and Katz, Schaps, and Vishne in dimension 2 and 3. As applications, we obtain explicit estimates for systolic genus of hyperbolic manifolds studied by Belolipetsky and the distance of homological codes constructed by Guth and Lubotzky. In Appendix A together with Cayo Dória we prove that the constant 8n(n+1)\frac{8}{n(n+1)} is sharp.

Cite this article

Plinio G.P. Murillo, Systole of congruence coverings of arithmetic hyperbolic manifolds. Groups Geom. Dyn. 13 (2019), no. 3, pp. 1083–1102

DOI 10.4171/GGD/515