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Systoles of hyperbolic hybrids

Sami Douba

Abstract

We exhibit closed hyperbolic manifolds with arbitrarily small systole in each dimension that are not quasi-arithmetic in the sense of Vinberg, and are thus not commensurable to those constructed by Agol, Belolipetsky--Thomson, and Bergeron--Haglund--Wise. This is done by taking hybrids of the manifolds constructed by the latter authors.

Systoles of hyperbolic hybrids

Abstract

We exhibit closed hyperbolic manifolds with arbitrarily small systole in each dimension that are not quasi-arithmetic in the sense of Vinberg, and are thus not commensurable to those constructed by Agol, Belolipetsky--Thomson, and Bergeron--Haglund--Wise. This is done by taking hybrids of the manifolds constructed by the latter authors.
Paper Structure (1 section, 2 theorems, 4 equations, 1 figure)

This paper contains 1 section, 2 theorems, 4 equations, 1 figure.

Table of Contents

  1. Acknowledgements

Key Result

Theorem 1

For each $n \geq 2$ and $\epsilon > 0$, there is a closed hyperbolic $n$-manifold of systole $< \epsilon$ that is not quasi-arithmetic.

Figures (1)

  • Figure 1: A schematic of the construction of the manifold $M$ in the proof of Theorem \ref{['main']}.

Theorems & Definitions (7)

  • Theorem 1
  • proof : Proof of Theorem \ref{['main']}
  • Remark 2
  • Remark 3
  • Theorem 4
  • proof
  • Remark 5