Systoles of Arithmetic Hyperbolic 2- and 3-Manifolds

Bozzai, Rainie; Linowitz, Benjamin

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 33 (2024) no. 4, pp. 997-1017

In this paper we study the systoles of arithmetic hyperbolic $2$- and $3$-manifolds. Our first result is the construction of infinitely many arithmetic hyperbolic $2$- and $3$-manifolds which are pairwise noncommensurable, all have the same systole, and whose volumes are explicitly bounded. Our second result fixes a positive number $x$ and gives an upper bound for the least volume of an arithmetic hyperbolic $2$- or $3$-manifold whose systole is greater than $x$. We conclude by determining, for certain small values of $x$, the least volume of a principal arithmetic hyperbolic $2$-manifold over $\mathbb{Q}$ or $3$-manifold over $\mathbb{Q}\left(i\right)$ whose systole is greater than $x$.

Dans cet article, nous étudions les systoles des variétés hyperboliques arithmétiques en dimension $2$ et $3$. Notre premier résultat est la construction d’une infinité de variétés hyperboliques arithmétiques qui sont incommensurables deux-à-deux, ont toutes la même systole, et dont les volumes sont explicitement bornés. Notre deuxième résultat suppose fixé $x>0$ et donne une borne supérieure pour le plus petit volume d’une variété hyperbolique arithmétique avec une systole supérieure à $x$. Nous concluons en déterminant, pour certaines petites valeurs de $x$, le plus petit volume d’une variété hyperbolique arithmétique principale sur $\mathbb{Q}$ (en dimension $2$) ou sur $\mathbb{Q}\left(i\right)$ (en dimension $3$) avec une systole supérieure à $x$.

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