Abstract
We investigate the existence and properties of uniform lattices in Lie groups and use these results to prove that, in dimension 5, there are exactly seven connected and simply connected contact Lie groups with uniform lattices, all of which are solvable. In particular, it is also shown that the special affine group has no uniform lattice.
Highlights
This paper investigates the geometry of compact contact manifolds that are uniformized by contact Lie groups, i.e., manifolds of the form Γ \ G for some Lie group G with a left invariant contact structure and uniform lattice Γ ⊂ G
We restrict our attention to dimension five and describe which five-dimensional contact Lie groups admit uniform lattices
Lie group if and only if there is a lattice ∆ of a -connected nilpotent Lie group and nonnegative integer k such that 0 → ∆ → Γ → Zk → 0 is a short, exact sequence. This implies that, if G = N ⋊b T is a -connected splittable solvable Lie group with nilradical N and Γ is a lattice of G, Γ is isomorphic to ∆ ⋊b TZ where ∆ is a lattice of N and TZ a lattice of T such that b(TZ) ⊂ Aut(∆)
Summary
This paper investigates the geometry of compact contact manifolds that are uniformized by contact Lie groups, i.e., manifolds of the form Γ \ G for some Lie group G with a left invariant contact structure and uniform lattice Γ ⊂ G. This includes both a review of several classical results and some original results regarding contact Lie groups. Constructs compact symplectic (2n + 2)-manifolds whose boundaries are disconnected contact (2n + 1)-manifolds uniformized by contact Lie groups and when n = 2, by the Lie groups of Theorem 3.1. This is a generalisation to all higher dimensions of a construction used in [12], to give counter-examples, when n = 1, to the question of E. This paper would not have been possible without this help
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