Systole of congruence coverings of arithmetic hyperbolic manifolds

Abstract

In this paper we prove that, for any arithmetic hyperbolic n-manifold M of the first type, the systole of most of the principal congruence coverings M I satisfywhere c is a constant independent of I. 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 an appendix together with Cayo Dória we prove that the constant 8 n(n+1) is sharp.

On the systole growth in congruence quaternionic hyperbolic manifolds

“…Note that when n = 1, H 1 H is isometric to the four dimensional real hyperbolic space, and the constant 2 5 agrees with that of [15]. In Section 6 we generalize the argument of Dória and Murillo to prove that the constant 4 (n+1)(2n+3) is optimal; see Theorem 6.1.…”

“…They also proved that for compact arithmetic hyperbolic 3-manifolds the constant C = 2 3 works. For real arithmetic hyperbolic n-manifold of the first type, these results were generalized by Murillo in [15], where he proved that the constant is equal to Recently, Lapan, Linowitz and Meyer obtained a value for the constant C for a large class of arithmetic locally symmetric spaces, including real, complex and quaternionic hyperbolic orbifolds [9]. However, the values of the constants obtained in [9] are not optimal, as the comparison with the results mentioned above shows.…”

Negatively curved Einstein metrics on ramified covers of closed four-dimensional hyperbolic manifolds

“…For us, of course, the interest is in n 4. This was treated in a recent article by Murillo [20], which shows that, with a couple of caveats, the optimal inequality has…”

“…We review this in the next Subsection 2.1. To bound the volume of Σ k we will make use of recent work of Murillo [20], which does not apply to all the sequences arising from Gromov-Thurston's original construction, but instead to a special subclass of them. Put loosely, we need the manifolds M k to be spin.…”

Examples of compact Einstein four-manifolds with negative curvature

“…For us, of course, the interest is in n ≥ 4. This was treated in a recent article by Murillo [29], which shows that, with a couple of caveats, the optimal inequality has C(n) = n(n + 1) 4…”