Talk:Lattice (discrete subgroup)

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Assessment comment[edit]

The comment(s) below were originally left at Talk:Lattice (discrete subgroup)/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

Hard to assign a unique field… Arcfrk (talk) 10:46, 16 February 2008 (UTC)[reply]

Last edited at 10:46, 16 February 2008 (UTC). Substituted at 21:42, 29 April 2016 (UTC)

Seems a one-sided viewpoint[edit]

"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)[reply]

Quasi-isometry and coarse equivalence[edit]

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)[reply]

Is "cocompact" a synonym of "uniform"?[edit]

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)[reply]

yes, done. jraimbau (talk) 06:00, 13 May 2022 (UTC)[reply]

Unclear terminology[edit]

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)[reply]

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)[reply]