Math 2.0

Topological groups, Lie groups Definition

Lattice in a locally compact group

Builds on Locally compact groupDiscrete subgroupHaar measureRadon measureUnimodular groupFundamental domain

Definition

Let GG be a locally compact group and let Γ\Gamma be a discrete subgroup of GG. Let G/ΓG/\Gamma be the space of left cosets gΓg\Gamma with the quotient topology, on which GG acts by h⋅gΓ=hgΓh \cdot g\Gamma = hg\Gamma. The subgroup Γ\Gamma is a lattice in GG if there is a nonzero Radon measure ν\nu on G/ΓG/\Gamma such that:

  1. ν\nu is GG-invariant: ν(hE)=ν(E)\nu(hE) = \nu(E) for every h∈Gh \in G and every Borel set E⊆G/ΓE \subseteq G/\Gamma;
  2. ν\nu is finite: ν(G/Γ)<∞\nu(G/\Gamma) < \infty.

A lattice Γ\Gamma is uniform, or cocompact, if G/ΓG/\Gamma is compact.

Remarks

Example: ℤn\Z^n in (ℝn,+)(\R^n, +). The quotient ℝn/ℤn\R^n/\Z^n is the torus, and the image of Lebesgue measure on [0,1)n[0, 1)^n is a translation-invariant measure on it of total mass 11. Since the torus is compact, ℤn\Z^n is a uniform lattice.

Non-example: Γ=ℤ×{0}\Gamma = \Z \times \set{0} in ℝ2\R^2 is discrete, but fails clause 2. The quotient is the cylinder (ℝ/ℤ)×ℝ(\R/\Z) \times \R. Every invariant Radon measure on it is a multiple of the product of the Haar measures of the two factors. The nonzero ones have infinite total mass.

If GG has a lattice, then GG is unimodular. A measurable fundamental domain for the action of Γ\Gamma on GG by right multiplication has finite Haar measure exactly when Γ\Gamma is a lattice. The group SL2(ℤ)SL_2(\Z) is a lattice in SL2(ℝ)SL_2(\R) that is not uniform.

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To understand this page from the ground up, read these 7 pages in order. The foundations come first.

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Discrete subgroup, concept, no evidence levelDiscrete subgroupconceptFundamental domain, concept, no evidence levelFundamental domainconceptHaar measure, concept, no evidence levelHaar measureconceptLocally compact group, concept, no evidence levelLocally compact groupconceptRadon measure, concept, no evidence levelRadon measureconceptUnimodular group, concept, no evidence levelUnimodular groupconceptLattice in a locally compact group, definition, no evidence levelLattice in a locally …definition
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  1. Discrete subgroup concept
  2. Fundamental domain concept
  3. Haar measure concept
  4. Locally compact group concept
  5. Radon measure concept
  6. Unimodular group concept
  7. Lattice in a locally compact group definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Lattice_(discrete_subgroup)
About this page
TagBW21VJW403
URIm2://BW21VJW403
Kinddefinition
WritingW0 stub
MSC22D05, 22E40, 28C10
DefinesLattice in a locally compact group
UsesLocally compact group, Discrete subgroup, Haar measure, Radon measure, Unimodular group, Fundamental domain

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