# Thin part of the arithmetic orbifolds

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Speaker:
Mikolaj Fraczyk
Affiliation:
University of Chicago
Date:
Mon, 16/05/2022 - 13:30 - 14:30

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.

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