Bottom of the Length Spectrum of Arithmetic Orbifolds

Mikolaj Fraczyk; Lam L. Pham

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arXiv · 2207.02332

Bottom of the Length Spectrum of Arithmetic Orbifolds

Mikolaj Fraczyk·Lam L. Pham

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Abstract

We prove that cocompact arithmetic lattices in a simple Lie group are uniformly discrete if and only if the Salem numbers are uniformly bounded away from $1$. We also prove an analogous result for semisimple Lie groups. Finally, we shed some light on the structure of the bottom of the length spectrum of an arithmetic orbifold $\Gamma\backslash X$ by showing the existence of a positive constant $\delta(X)>0$ such that squares of lengths of closed geodesics shorter than $\delta$ must be pairwise linearly dependent over $\mathbb Q$.

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BibTeX RIS

Mikolaj Fraczyk, Lam L. Pham. 2022-07-05. Bottom of the Length Spectrum of Arithmetic Orbifolds. https://arxiv.org/abs/2207.02332

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