A. We generalize a result of Sury [Sur92] and prove that uniform discreteness of cocompact lattices in higher rank semisimple Lie groups (first conjectured by Margulis [Mar91]) is equivalent to a weak form of Lehmer's conjecture. We include a short survey of related results and conjectures.
A. 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 Γ\ by showing the existence of a positive constant ( ) > 0 such that squares of lengths of closed geodesics shorter than must be pairwise linearly dependent over Q.
Let S ⊂ GL n ( Z ) S\subset\mathrm{GL}_{n}(\mathbb{Z}) be a finite symmetric set. We show that if the Zariski closure of Γ = ⟨ S ⟩ \Gamma=\langle S\rangle is a product of special linear groups or a special affine linear group, then the diameter of the Cayley graph Cay ( Γ / Γ ( q ) , π q ( S ) ) \operatorname{Cay}(\Gamma/\Gamma(q),\pi_{q}(S)) is O ( log q ) O(\log q) , where 𝑞 is an arbitrary positive integer, π q : Γ → Γ / Γ ( q ) \pi_{q}\colon\Gamma\to\Gamma/\Gamma(q) is the canonical projection induced by the reduction modulo 𝑞, and the implied constant depends only on 𝑆.
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