Talk:Lattice (discrete subgroup)

Assessment comment
Substituted at 21:42, 29 April 2016 (UTC)

Seems a one-sided viewpoint
"Lattices are best thought of as discrete approximations of continuous groups (such as Lie groups)."

This isn't the best viewpoint if your interest is the quotient space.

188.154.206.128 (talk) 16:15, 21 January 2019 (UTC)

Quasi-isometry and coarse equivalence
I'm not sure whether discussion of these notions is relevant in the introductory section. As far as i can tell it is equivalent for a discrete subgroup to be either a uniform lattice, quasi-isometric to or coarsely equivalent to its ambient group (with an invariant metric), though i don't know any reference for the latter.

It could make sense to add a section about "lattices in geometric group theory" or something similar where this is discussed. jraimbau (talk) 12:11, 27 October 2021 (UTC)

Is "cocompact" a synonym of "uniform"?
In the section Generalities on lattices this sentence appears:

"A lattice $$\Gamma \subset G$$ is called uniform when the quotient space $$G/\Gamma$$ is compact (and non-uniform otherwise)."

Am I correct to say that, when speaking of a lattice in a Lie group, "cocompact" is a synonym for "uniform"?

If so, then this is worth mentioning in the artice. 2601:200:C000:1A0:C0A2:E29D:72EF:28D2 (talk) 19:10, 12 May 2022 (UTC)


 * yes, done. jraimbau (talk) 06:00, 13 May 2022 (UTC)

Unclear terminology
The section "Rank 1 versus higher rank''' begins with this sentence:

"The real rank of a Lie group is the maximal dimension of an abelian subgroup containing only semisimple elements.'

The linked article Semisimple does not explain what a "semisimple element" is.

I hope someone knowledgeable about this subject can clarify this. 2601:200:C000:1A0:C0A2:E29D:72EF:28D2 (talk) 19:20, 12 May 2022 (UTC)


 * The article on semisimple operators does define what it means pretty clearly (an element of a Lie group is a linear operator via a faithful linear representation of the Lie group, for instance the adjoint representation, and semisimplicity does not depend on the representation). jraimbau (talk) 05:06, 13 May 2022 (UTC)