- Lattice (discrete subgroup)
In

Lie theory and related areas ofmathematics , a**lattice**in alocally compact topological group is adiscrete subgroup with the property that the quotient space has finite invariant measure. In the special case of subgroups of**R**^{"n"}, this amounts to the usual geometric notion of a lattice, and both the algebraic structure of lattices and the geometry of the totality of all lattices are relatively well understood. Deep results of Borel,Harish-Chandra , Mostow, Tamagawa,M.S. Raghunathan , Margulis, Zimmer obtained from the 1950s through the 1970s provided examples and generalized much of the theory to the setting of nilpotentLie group s andsemisimple algebraic group s over alocal field . In the 1990s, Bass and Lubotzky initiated the study of "tree lattices", which remains an active research area.**Definition**Let "G" be a locally compact topological group with the

Haar measure "μ". A discrete subgroup "Γ" is called a**lattice**in "G" if the quotient space "G"/"Γ" has finite invariant measure, that is, if "G" is a unimodular group and the volume "μ"("G"/"Γ") is finite. The lattice is**uniform**(or "cocompact") if the quotient space iscompact , and**nonuniform**otherwise.**Arithmetic lattices**An archetypical example of a nonuniform lattice is given by the group "SL"(2,

**Z**), which is a lattice in thespecial linear group "SL"(2,**R**), and by the closely relatedmodular group . This construction admits a far-reaching generalization to a class of lattices in allsemisimple algebraic group s over alocal field "F" called "arithmetic lattices". For example, let "F" =**R**be the field of real numbers. Roughly speaking, theLie group "G"(**R**) is formed by all matrices with entries in**R**satisfying certain algebraic conditions, and by restricting the entries to theinteger s**Z**, one obtains a lattice "G"(**Z**). Conversely,Grigory Margulis proved that under certain assumptions on "G", any lattice in it essentially arises in this way. This remarkable statement is known as "Arithmeticity of lattices" or "Margulis Arithmeticity Theorem".**"S"-arithmetic lattices**Arithmetic lattices admit an important generalization, known as the "S"-"arithmetic lattices". The first example is given by the diagonally embedded subgroup

: $SLleft(2,mathbb\{Z\}left\; [frac\{1\}\{p\}\; ight]\; ight)\; subset\; SL(2,mathbb\{R\})\; imes\; SL(2,mathbb\{Q\}\_p),\; S=\{p,\; infty\}.$

This is a lattice in the product of algebraic groups over "different" local fields, both real and p-adic. It is formed by the unimodular matrices of order 2 with entries in the localization of the ring of integers at the prime "p". The set "S" is a finite set of places of

**Q**which includes all archimedean places and the locally compact group is the direct product of the groups of points of a fixed linear algebraic group "G" defined over**Q**(or a more generalglobal field ) over the completions of**Q**at the places from "S". To form the discrete subgroup, instead of matrices with integer entries, one considers matrices with entries in the localization over the primes (nonarchimedean places) in "S". Under fairly general assumptions, this construction indeed produces a lattice. The class of "S"-arithmetic lattices is much wider than the class of arithmetic lattices, but they share many common features.**Adelic case**A lattice of fundamental importance for the theory of

automorphic form s is given by the group "G"("K") of "K"-points of a semisimple (or reductive) linear algebraic group "G" defined over a global field "K". This group diagonally embeds into theadelic algebraic group "G"("A"), where "A" is the ring of adeles of "K", and is a lattice there. Unlike arithmetic lattices, "G"("K") is not finitely generated.**Rigidity**Another group of phenomena concerning lattices in semisimple algebraic groups is collectively known as "rigidity".

Mostow rigidity theorem showed that the algebraic structure of a lattice in simple Lie group "G" of split rank at least two determines "G". Thus any isomorphism of lattices in two such groups is essentially induced by an isomorphism between the groups themselves. "Superrigidity" provides a generalization dealing with homomorphisms from a lattice in an algebraic group "G" into another algebraic group "H".**Tree lattices**Let "X" be a locally finite tree. Then the

automorphism group "G" of "X" is a locally compact topological group, in which the basis of the topology is given by the stabilizers of finite sets of vertices. Vertex stabilizers "G"_{"x"}are thus compact open subgroups, and a subgroup "Γ" of "G" is discrete if "Γ"_{"x"}is finite for some (and hence, for any) vertex "x". The subgroup "Γ" is an**"X"-lattice**if the suitably defined volume of $Gammaackslash\; X$ is finite, and a**uniform "X"-lattice**if this quotient is a finite graph. In case $Gackslash\; X$ is finite, this is equivalent to "Γ" being a lattice (respectively, a uniform lattice) in "G".**See also***

Kazhdan's property (T)

*Graph of groups **References***

Hyman Bass and Alexander Lubotzky, "Tree lattices". With appendices by H. Bass, L. Carbone, A. Lubotzky, G. Rosenberg, and J. Tits. Progress in Mathematics, vol 176, Birkhäuser Verlag, Boston, 2001 ISBN 0-8176-4120-3

*Grigory Margulis , "Discrete subgroups of semisimple Lie groups",Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)] , 17.Springer-Verlag , Berlin, 1991. x+388 pp. ISBN 3-540-12179-X MathSciNet|id=1090825

* [*http://people.uleth.ca/~dave.morris/LectureNotes.shtml#ArithmeticGroups Dave Witte Morris: Introduction to Arithmetic Groups*] , draft of a book

*citation|id=MR|1278263

last=Platonov|first= Vladimir|author-link=Vladimir Platonov|last2= Rapinchuk|first2= Andrei|title=Algebraic groups and number theory. (Translated from the 1991 Russian original by Rachel Rowen.) |series=Pure and Applied Mathematics|volume= 139|publisher= Academic Press, Inc.|publication-place= Boston, MA|year= 1994|ISBN= 0-12-558180-7

* M.S.Raghunathan, "Discrete subgroups of Lie groups".Ergebnisse der Mathematik und ihrer Grenzgebiete , Band 68.Springer-Verlag , New York-Heidelberg, 1972 MathSciNet|id=0507234

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