Lectures on Lattices

Let G be a locally compact group equipped with a left Haar measure μ, i.e. a Borel regular measure which is finite on compact, positive on open and invariant under left multiplications — by Haar’s theorem such μ exists and is unique up to normalization. The group G is called unimodular if μ is also right invariant, or equivalently if it is symmetric in the sense that μ(A) = μ(A−1) for every measurable set A. Note that G is compact iff μ(G) <∞. For example: • Compact groups, Nilpotent groups and Perfect groups are unimodular. • The group of affine transformations of the real line is not unimodular. A closed subgroup H ≤ G is said to be co-finite if the quotient space G/H admits a non-trivial finite G invariant measure. A lattice in G is a co-finite discrete subgroup. A discrete subgroup Γ ≤ G is a lattice iff it admits a finite measure fundamental domain, i.e. a measurable set Ω of finite measure which form a set of right cosets representatives for Γ in G. We shall denote Γ ≤L G to express that Γ is a lattice in G.