Thin part of the arithmetic orbifolds

Virtual talk.

Let \(X\) be a symmetric space. The collar lemma, also known as the Margulis lemma, says that there exists an \(\varepsilon=\varepsilon(X)\) such that the \(\varepsilon\)-thin part of a locally symmetric space \(X/ \Gamma\) looks locally like a quotient by a virtually unipotent subgroup. It turns out that in the arithmetic setting we can improve this lemma by making \(\varepsilon\) grow linearly in the degree of the number filed generated by the traces of elements of \(\Gamma\). I will explain why this is the case and present several applications, including the proof of the fact that an arithmetic locally symmetric manifold \(M\) is homotopy equivalent to a simplicial complex of size bounded linearly in the volume of \(M\) and degrees of all vertices bounded uniformly in terms of \(X\). Based on a joint work with Sebastian Hurtado and Jean Raimbault.