Arithmetic Lattice
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In
Lie theory In mathematics, the mathematician Sophus Lie ( ) initiated lines of study involving integration of differential equations, transformation groups, and contact of spheres that have come to be called Lie theory. For instance, the latter subject is L ...
and related areas of mathematics, a lattice in a
locally compact group In mathematics, a locally compact group is a topological group ''G'' for which the underlying topology is locally compact and Hausdorff. Locally compact groups are important because many examples of groups that arise throughout mathematics are lo ...
is a
discrete subgroup In mathematics, a topological group ''G'' is called a discrete group if there is no limit point in it (i.e., for each element in ''G'', there is a neighborhood which only contains that element). Equivalently, the group ''G'' is discrete if and o ...
with the property that the quotient space has finite
invariant measure In mathematics, an invariant measure is a measure that is preserved by some function. The function may be a geometric transformation. For examples, circular angle is invariant under rotation, hyperbolic angle is invariant under squeeze mapping, an ...
. In the special case of subgroups of R''n'', this amounts to the usual geometric notion of a lattice as a periodic subset of points, and both the algebraic structure of lattices and the geometry of the space of all lattices are relatively well understood. The theory is particularly rich for lattices in semisimple Lie groups or more generally in
semisimple algebraic group In mathematics, a reductive group is a type of linear algebraic group over a field. One definition is that a connected linear algebraic group ''G'' over a perfect field is reductive if it has a representation with finite kernel which is a direc ...
s over
local field In mathematics, a field ''K'' is called a (non-Archimedean) local field if it is complete with respect to a topology induced by a discrete valuation ''v'' and if its residue field ''k'' is finite. Equivalently, a local field is a locally compact t ...
s. In particular there is a wealth of rigidity results in this setting, and a celebrated theorem of
Grigory Margulis Grigory Aleksandrovich Margulis (russian: Григо́рий Алекса́ндрович Маргу́лис, first name often given as Gregory, Grigori or Gregori; born February 24, 1946) is a Russian-American mathematician known for his work on ...
states that in most cases all lattices are obtained as
arithmetic group In mathematics, an arithmetic group is a group obtained as the integer points of an algebraic group, for example \mathrm_2(\Z). They arise naturally in the study of arithmetic properties of quadratic forms and other classical topics in number theor ...
s. Lattices are also well-studied in some other classes of groups, in particular groups associated to
Kac–Moody algebra In mathematics, a Kac–Moody algebra (named for Victor Kac and Robert Moody, who independently and simultaneously discovered them in 1968) is a Lie algebra, usually infinite-dimensional, that can be defined by generators and relations through a ge ...
s and automorphisms groups of regular
trees In botany, a tree is a perennial plant with an elongated stem, or trunk, usually supporting branches and leaves. In some usages, the definition of a tree may be narrower, including only woody plants with secondary growth, plants that are u ...
(the latter are known as ''tree lattices''). Lattices are of interest in many areas of mathematics:
geometric group theory Geometric group theory is an area in mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such group (mathematics), groups and topology, topological and geometry, geometric pro ...
(as particularly nice examples of
discrete group In mathematics, a topological group ''G'' is called a discrete group if there is no limit point in it (i.e., for each element in ''G'', there is a neighborhood which only contains that element). Equivalently, the group ''G'' is discrete if and on ...
s), in
differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...
(through the construction of locally homogeneous manifolds), in number theory (through
arithmetic group In mathematics, an arithmetic group is a group obtained as the integer points of an algebraic group, for example \mathrm_2(\Z). They arise naturally in the study of arithmetic properties of quadratic forms and other classical topics in number theor ...
s), in
ergodic theory Ergodic theory (Greek: ' "work", ' "way") is a branch of mathematics that studies statistical properties of deterministic dynamical systems; it is the study of ergodicity. In this context, statistical properties means properties which are expres ...
(through the study of homogeneous flows on the quotient spaces) and in
combinatorics Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many appl ...
(through the construction of expanding
Cayley graph In mathematics, a Cayley graph, also known as a Cayley color graph, Cayley diagram, group diagram, or color group is a graph that encodes the abstract structure of a group. Its definition is suggested by Cayley's theorem (named after Arthur Cayle ...
s and other combinatorial objects).


Generalities on lattices


Informal discussion

Lattices are best thought of as discrete approximations of continuous groups (such as Lie groups). For example, it is intuitively clear that the subgroup \mathbb Z^n of integer vectors "looks like" the real vector space \mathbb R^n in some sense, while both groups are essentially different: one is finitely generated and
countable In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers; ...
, while the other is not finitely generated and has the
cardinality of the continuum In set theory, the cardinality of the continuum is the cardinality or "size" of the set of real numbers \mathbb R, sometimes called the continuum. It is an infinite cardinal number and is denoted by \mathfrak c (lowercase fraktur "c") or , \mathb ...
. Rigorously defining the meaning of "approximation of a continuous group by a discrete subgroup" in the previous paragraph in order to get a notion generalising the example \mathbb Z^n \subset \mathbb R^n is a matter of what it is designed to achieve. Maybe the most obvious idea is to say that a subgroup "approximates" a larger group is that the larger group can be covered by the translates of a "small" subset by all elements in the subgroups. In a locally compact topological group there are two immediately available notions of "small": topological (a
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...
, or
relatively compact subset In mathematics, a relatively compact subspace (or relatively compact subset, or precompact subset) of a topological space is a subset whose closure is compact. Properties Every subset of a compact topological space is relatively compact (sinc ...
) or measure-theoretical (a subset of finite Haar measure). Note that since the Haar measure is a
Radon measure In mathematics (specifically in measure theory), a Radon measure, named after Johann Radon, is a measure on the σ-algebra of Borel sets of a Hausdorff topological space ''X'' that is finite on all compact sets, outer regular on all Borel se ...
, so it gives finite mass to compact subsets, the second definition is more general. The definition of a lattice used in mathematics relies upon the second meaning (in particular to include such examples as \mathrm_2(\mathbb Z) \subset \mathrm_2(\mathbb R)) but the first also has its own interest (such lattices are called uniform). Other notions are coarse equivalence and the stronger
quasi-isometry In mathematics, a quasi-isometry is a function between two metric spaces that respects large-scale geometry of these spaces and ignores their small-scale details. Two metric spaces are quasi-isometric if there exists a quasi-isometry between them. T ...
. Uniform lattices are quasi-isometric to their ambient groups, but non-uniform ones are not even coarsely equivalent to it.


Definition

Let G be a locally compact group and \Gamma a discrete subgroup (this means that there exists a neighbourhood U of the identity element e_G of G such that \Gamma \cap U = \). Then \Gamma is called a lattice in G if in addition there exists a
Borel measure In mathematics, specifically in measure theory, a Borel measure on a topological space is a measure that is defined on all open sets (and thus on all Borel sets). Some authors require additional restrictions on the measure, as described below. F ...
\mu on the quotient space G / \Gamma which is finite (i.e. \mu(G / \Gamma) < +\infty) and G-invariant (meaning that for any g \in G and any open subset W \subset G / \Gamma the equality \mu(gW) = \mu(W) is satisfied). A slightly more sophisticated formulation is as follows: suppose in addition that G is unimodular, then since \Gamma is discrete it is also unimodular and by general theorems there exists a unique G-invariant Borel measure on G / \Gamma up to scaling. Then \Gamma is a lattice if and only if this measure is finite. In the case of discrete subgroups this invariant measure coincides locally with the
Haar measure In mathematical analysis, the Haar measure assigns an "invariant volume" to subsets of locally compact topological groups, consequently defining an integral for functions on those groups. This measure was introduced by Alfréd Haar in 1933, though ...
and hence a discrete subgroup in a locally compact group G being a lattice is equivalent to it having a fundamental domain (for the action on G by left-translations) of finite volume for the Haar measure. A lattice \Gamma \subset G is called uniform (or cocompact) when the quotient space G/\Gamma is compact (and ''non-uniform'' otherwise). Equivalently a discrete subgroup \Gamma \subset G is a uniform lattice if and only if there exists a compact subset C \subset G with G = \bigcup _ \, C\gamma. Note that if \Gamma is any discrete subgroup in G such that G/\Gamma is compact then \Gamma is automatically a lattice in G.


First examples

The fundamental, and simplest, example is the subgroup \mathbb Z^n which is a lattice in the Lie group \mathbb R^n. A slightly more complicated example is given by the discrete
Heisenberg group In mathematics, the Heisenberg group H, named after Werner Heisenberg, is the group of 3×3 upper triangular matrices of the form ::\begin 1 & a & c\\ 0 & 1 & b\\ 0 & 0 & 1\\ \end under the operation of matrix multiplication. Elements ' ...
inside the continuous Heisenberg group. If G is a discrete group then a lattice in G is exactly a subgroup \Gamma of finite index (i.e. the quotient set G/\Gamma is finite). All of these examples are uniform. A non-uniform example is given by the
modular group In mathematics, the modular group is the projective special linear group of matrices with integer coefficients and determinant 1. The matrices and are identified. The modular group acts on the upper-half of the complex plane by fractional l ...
\mathrm_2(\mathbb Z) inside \mathrm_2(\mathbb R), and also by the higher-dimensional analogues \mathrm_n(\mathbb Z) \subset \mathrm_n(\mathbb R). Any finite-index subgroup of a lattice is also a lattice in the same group. More generally, a subgroup commensurable to a lattice is a lattice.


Which groups have lattices?

Not every locally compact group contains a lattice, and there is no general group-theoretical sufficient condition for this. On the other hand, there are plenty of more specific settings where such criteria exist. For example, the existence or non-existence of lattices in
Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...
s is a well-understood topic. As we mentioned, a necessary condition for a group to contain a lattice is that the group must be unimodular. This allows for the easy construction of groups without lattices, for example the group of invertible
upper triangular matrices In mathematics, a triangular matrix is a special kind of square matrix. A square matrix is called if all the entries ''above'' the main diagonal are zero. Similarly, a square matrix is called if all the entries ''below'' the main diagonal are ...
or the
affine group In mathematics, the affine group or general affine group of any affine space over a field is the group of all invertible affine transformations from the space into itself. It is a Lie group if is the real or complex field or quaternions. Relat ...
s. It is also not very hard to find unimodular groups without lattices, for example certain nilpotent Lie groups as explained below. A stronger condition than unimodularity is
simplicity Simplicity is the state or quality of being simple. Something easy to understand or explain seems simple, in contrast to something complicated. Alternatively, as Herbert A. Simon suggests, something is simple or complex depending on the way we ch ...
. This is sufficient to imply the existence of a lattice in a Lie group, but in the more general setting of locally compact groups there exists simple groups without lattices, for example the "Neretin groups".


Lattices in solvable Lie groups


Nilpotent Lie groups

For nilpotent groups the theory simplifies much from the general case, and stays similar to the case of Abelian groups. All lattices in a nilpotent Lie group are uniform, and if N is a connected
simply connected In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the spac ...
nilpotent Lie group (equivalently it does not contain a nontrivial compact subgroup) then a discrete subgroup is a lattice if and only if it is not contained in a proper connected subgroup (this generalises the fact that a discrete subgroup in a vector space is a lattice if and only if it spans the vector space). A nilpotent Lie group ''G'' contains a lattice if and only if the Lie algebra 𝓰 of ''G'' can be defined over the rationals. That is, if and only if the
structure constants In mathematics, the structure constants or structure coefficients of an algebra over a field are used to explicitly specify the product of two basis vectors in the algebra as a linear combination. Given the structure constants, the resulting prod ...
of 𝓰 are rational numbers. More precisely: In a nilpotent group whose Lie algebra has only rational structure constants, lattices are the images via the exponential map of lattices (in the more elementary sense of
Lattice (group) In geometry and group theory, a lattice in the real coordinate space \mathbb^n is an infinite set of points in this space with the properties that coordinate wise addition or subtraction of two points in the lattice produces another lattice poi ...
) in the Lie algebra. A lattice in a nilpotent Lie group N is always finitely generated (and hence finitely presented since it is itself nilpotent); in fact it is generated by at most \dim(G) elements. Finally, a nilpotent group is isomorphic to a lattice in a nilpotent Lie group if and only if it contains a subgroup of finite index which is torsion-free and finitely generated.


The general case

The criterion for nilpotent Lie groups to have a lattice given above does not apply to more general solvable Lie groups. It remains true that any lattice in a solvable Lie group is uniform and that lattices in solvable groups are finitely presented. Not all finitely generated solvable groups are lattices in a Lie group. An algebraic criterion is that the group be polycyclic.


Lattices in semisimple Lie groups


Arithmetic groups and existence of lattices

If G is a semisimple
linear algebraic group In mathematics, a linear algebraic group is a subgroup of the group of invertible n\times n matrices (under matrix multiplication) that is defined by polynomial equations. An example is the orthogonal group, defined by the relation M^TM = I_n wh ...
in \mathrm_n(\mathbb R) which is defined over the field \mathbb Q of
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ration ...
s (i.e. the polynomial equations defining G have their coefficients in \mathbb Q) then it has a subgroup \Gamma = G \cap \mathrm_n(\mathbb Z). A fundamental theorem of
Armand Borel Armand Borel (21 May 1923 – 11 August 2003) was a Swiss mathematician, born in La Chaux-de-Fonds, and was a permanent professor at the Institute for Advanced Study in Princeton, New Jersey, United States from 1957 to 1993. He worked in alg ...
and
Harish-Chandra Harish-Chandra Fellow of the Royal Society, FRS (11 October 1923 – 16 October 1983) was an Indian American mathematician and physicist who did fundamental work in representation theory, especially harmonic analysis on semisimple Lie groups. ...
states that \Gamma is always a lattice in G; the simplest example of this is the subgroup \mathrm_2(\mathbb Z) \subset \mathrm_2(\mathbb R). Generalising the construction above one gets the notion of an ''arithmetic lattice'' in a semisimple Lie group. Since all semisimple Lie groups can be defined over \mathbb Q a consequence of the arithmetic construction is that any semisimple Lie group contains a lattice.


Irreducibility

When the Lie group G splits as a product G = G_1 \times G_2 there is an obvious construction of lattices in G from the smaller groups: if \Gamma_1 \subset G_1, \Gamma_2 \subset G_2 are lattices then \Gamma_1 \times \Gamma_2 \subset G is a lattice as well. Roughly, a lattice is then said to be ''irreducible'' if it does not come from this construction. More formally, if G = G_1 \times \ldots \times G_r is the decomposition of G into simple factors, a lattice \Gamma \subset G is said to be irreducible if either of the following equivalent conditions hold: *The projection of \Gamma to any factor G_ \times \ldots \times G_ is dense; *The intersection of \Gamma with any factor G_ \times \ldots \times G_ is not a lattice. An example of an irreducible lattice is given by the subgroup \mathrm_2(\mathbb Z sqrt 2 which we view as a subgroup \mathrm_2(\mathbb R) \times \mathrm_2(\mathbb R) via the map g \mapsto (g, \sigma(g)) where \sigma is the Galois map sending a matric with coefficients a_i+b_i\sqrt 2 to a_i - b_i \sqrt 2.


Rank 1 versus higher rank

The real rank of a Lie group G is the maximal dimension of a \mathbb R-split
torus In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle. If the axis of revolution does not tou ...
of G (an abelian subgroup containing only
semisimple In mathematics, semi-simplicity is a widespread concept in disciplines such as linear algebra, abstract algebra, representation theory, category theory, and algebraic geometry. A semi-simple object is one that can be decomposed into a sum of ''sim ...
elements with at least on real eigenvalue distinct from \pm 1). The semisimple Lie groups of real rank 1 without compact factors are (up to
isogeny In mathematics, in particular, in algebraic geometry, an isogeny is a morphism of algebraic groups (also known as group varieties) that is surjective and has a finite kernel. If the groups are abelian varieties, then any morphism of the underlying ...
) those in the following list (see
List of simple Lie groups In mathematics, a simple Lie group is a connected non-abelian Lie group ''G'' which does not have nontrivial connected normal subgroups. The list of simple Lie groups can be used to read off the list of simple Lie algebras and Riemannian symme ...
): *The
orthogonal group In mathematics, the orthogonal group in dimension , denoted , is the Group (mathematics), group of isometry, distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by ...
s \mathrm(n,1) of real quadratic forms of signature (n, 1) for n \ge 2; *The
unitary group In mathematics, the unitary group of degree ''n'', denoted U(''n''), is the group of unitary matrices, with the group operation of matrix multiplication. The unitary group is a subgroup of the general linear group . Hyperorthogonal group is an ...
s \mathrm(n,1) of
Hermitian form In mathematics, a sesquilinear form is a generalization of a bilinear form that, in turn, is a generalization of the concept of the dot product of Euclidean space. A bilinear form is linear in each of its arguments, but a sesquilinear form allows o ...
s of signature (n, 1) for n \ge 2; *The groups \mathrm(n,1) (groups of matrices with
quaternion In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quatern ...
coefficients which preserve a "quaternionic quadratic form" of signature (n, 1)) for n \ge 2; *The
exceptional Lie group In mathematics, a simple Lie group is a connected non-abelian Lie group ''G'' which does not have nontrivial connected normal subgroups. The list of simple Lie groups can be used to read off the list of simple Lie algebras and Riemannian symm ...
F_4^ (the real form of rank 1 corresponding to the exceptional Lie algebra F_4). The real rank of a Lie group has a significant influence on the behaviour of the lattices it contains. In particular the behaviour of lattices in the first two families of groups (and to a lesser extent that of lattices in the latter two) differs much from that of irreducible lattices in groups of higher rank. For example: *There exists non-arithmetic lattices in all groups \mathrm(n,1), in \mathrm(2,1),\mathrm(3,1), and possibly in \mathrm(n,1), n \ge 4 (the last is an open question) but all irreducible lattices in the others are arithmetic; *Lattices in rank 1 Lie groups have infinite, infinite index
normal subgroup In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group G i ...
s while all normal subgroups of irreducible lattices in higher rank are either of finite index or contained in their center; *Conjecturally, arithmetic lattices in higher-rank groups have the congruence subgroup property but there are many lattices in \mathrm(n,1), \mathrm(n,1) which have non-congruence finite-index subgroups.


Kazhdan's property (T)

The property known as (T) was introduced by Kazhdan to study the algebraic structure lattices in certain Lie groups when the classical, more geometric methods failed or at least were not as efficient. The fundamental result when studying lattices is the following: :''A lattice in a locally compact group has property (T) if and only if the group itself has property (T). '' Using
harmonic analysis Harmonic analysis is a branch of mathematics concerned with the representation of Function (mathematics), functions or signals as the Superposition principle, superposition of basic waves, and the study of and generalization of the notions of Fo ...
it is possible to classify semisimple Lie groups according to whether or not they have the property. As a consequence we get the following result, further illustrating the dichotomy of the previous section: *Lattices in \mathrm(n,1), \mathrm(n,1) do not have Kazhdan's property (T) while irreducible lattices in all other simple Lie groups do;


Finiteness properties

Lattices in semisimple Lie groups are always finitely presented, and actually satisfy stronger finiteness conditions. For uniform lattices this is a direct consequence of cocompactness. In the non-uniform case this can be proved using reduction theory. It is easier to prove finite presentability for groups with Property (T); however, there is a geometric proof which works for all semisimple Lie groups.


Riemannian manifolds associated to lattices in Lie groups


Left-invariant metrics

If G is a Lie group then from an
inner product In mathematics, an inner product space (or, rarely, a Hausdorff space, Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation (mathematics), operation called an inner product. The inner product of two ve ...
g_e on the tangent space \mathfrak g (the Lie algebra of G) one can construct a
Riemannian metric In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space ''T ...
on G as follows: if v, w belong to the tangent space at a point \gamma \in G put g_\gamma(v, w) = g_e(\gamma^*v, \gamma^*w) where \gamma^* indicates the
tangent map In differential geometry, pushforward is a linear approximation of smooth maps on tangent spaces. Suppose that is a smooth map between smooth manifolds; then the differential of ''φ, d\varphi_x,'' at a point ''x'' is, in some sense, the best ...
(at \gamma) of the diffeomorphism x \mapsto \gamma^x of G. The maps x \mapsto \gamma x for \gamma \in G are by definition isometries for this metric g. In particular, if \Gamma is any discrete subgroup in G (so that it acts freely and
properly discontinuously In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism g ...
by left-translations on G) the quotient \Gamma \backslash G is a Riemannian manifold locally isometric to G with the metric g. The
Riemannian volume form In mathematics, a volume form or top-dimensional form is a differential form of degree equal to the differentiable manifold dimension. Thus on a manifold M of dimension n, a volume form is an n-form. It is an element of the space of sections of the ...
associated to g defines a Haar measure on G and we see that the quotient manifold is of finite Riemannian volume if and only if \Gamma is a lattice. Interesting examples in this class of Riemannian spaces include compact
flat manifold In mathematics, a Riemannian manifold is said to be flat if its Riemann curvature tensor is everywhere zero. Intuitively, a flat manifold is one that "locally looks like" Euclidean space in terms of distances and angles, e.g. the interior angles o ...
s and
nilmanifold In mathematics, a nilmanifold is a differentiable manifold which has a transitive nilpotent group of diffeomorphisms acting on it. As such, a nilmanifold is an example of a homogeneous space and is diffeomorphic to the quotient space N/H, th ...
s.


Locally symmetric spaces

A natural inner product on \mathfrak g is given by the
Killing form In mathematics, the Killing form, named after Wilhelm Killing, is a symmetric bilinear form that plays a basic role in the theories of Lie groups and Lie algebras. Cartan's criteria (criterion of solvability and criterion of semisimplicity) show ...
. If G is not compact it is not definite and hence not an inner product: however when G is semisimple and K is a maximal compact subgroup it can be used to define a G-invariant metric on the
homogeneous space In mathematics, particularly in the theories of Lie groups, algebraic groups and topological groups, a homogeneous space for a group ''G'' is a non-empty manifold or topological space ''X'' on which ''G'' acts transitively. The elements of ' ...
X = G/K: such Riemannian manifolds are called symmetric spaces of non-compact type without Euclidean factors. A subgroup \Gamma \subset G acts freely, properly discontinuously on X if and only if it is discrete and torsion-free. The quotients \Gamma \backslash X are called locally symmetric spaces. There is thus a bijective correspondence between complete locally symmetric spaces locally isomorphic to X and of finite Riemannian volume, and torsion-free lattices in G. This correspondence can be extended to all lattices by adding
orbifold In the mathematical disciplines of topology and geometry, an orbifold (for "orbit-manifold") is a generalization of a manifold. Roughly speaking, an orbifold is a topological space which is locally a finite group quotient of a Euclidean space. D ...
s on the geometric side.


Lattices in p-adic Lie groups

A class of groups with similar properties (with respect to lattices) to real semisimple Lie groups are semisimple algebraic groups over local fields of characteristic 0, for example the
p-adic field In mathematics, the -adic number system for any prime number  extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems. The extension ...
s \mathbb Q_p. There is an arithmetic construction similar to the real case, and the dichotomy between higher rank and rank one also holds in this case, in a more marked form. Let G be an algebraic group over \mathbb Q_p of split-\mathbb Q_p-rank ''r''. Then: *If ''r'' is at least 2 all irreducible lattices in G are arithmetic; *if ''r=1'' then there are uncountably many commensurability classes of non-arithmetic lattices. In the latter case all lattices are in fact free groups (up to finite index).


S-arithmetic groups

More generally one can look at lattices in groups of the form :G = \prod_ G_p where G_p is a semisimple algebraic group over \mathbb Q_p. Usually p=\infty is allowed, in which case G_\infty is a real Lie group. An example of such a lattice is given by :\mathrm_2 \left( \mathbb Z \left frac 1 p \right\right) \subset \mathrm_2(\mathbb R) \times \mathrm_2(\mathbb Q_p). This arithmetic construction can be generalised to obtain the notion of an ''S-arithmetic group''. The Margulis arithmeticity theorem applies to this setting as well. In particular, if at least two of the factors G_p are noncompact then any irreducible lattice in G is S-arithmetic.


Lattices in adelic groups

If \mathrm G is a semisimple algebraic group over a
number field In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension). Thus K is a f ...
F and \mathbb A its adèle ring then the group G = \mathrm G(\mathbb A) of adélic points is well-defined (modulo some technicalities) and it is a locally compact group which naturally contains the group \mathrm G(F) of F-rational point as a discrete subgroup. The Borel–Harish-Chandra theorem extends to this setting, and \mathrm G(F) \subset \mathrm G(\mathbb A) is a lattice. The
strong approximation theorem In algebraic group theory, approximation theorems are an extension of the Chinese remainder theorem to algebraic groups ''G'' over global fields ''k''. History proved strong approximation for some classical groups. Strong approximation was establi ...
relates the quotient \mathrm G(F) \backslash \mathrm G(\mathbb A) to more classical S-arithmetic quotients. This fact makes the adèle groups very effective as tools in the theory of
automorphic form In harmonic analysis and number theory, an automorphic form is a well-behaved function from a topological group ''G'' to the complex numbers (or complex vector space) which is invariant under the action of a discrete subgroup \Gamma \subset G of ...
s. In particular modern forms of the trace formula are usually stated and proven for adélic groups rather than for Lie groups.


Rigidity


Rigidity results

Another group of phenomena concerning lattices in semisimple algebraic groups is collectively known as ''rigidity''. Here are three classical examples of results in this category.
Local rigidity Local rigidity theorems in the theory of discrete subgroups of Lie groups are results which show that small deformations of certain such subgroups are always trivial. It is different from Mostow rigidity and weaker (but holds more frequently) than ...
results state that in most situations every subgroup which is sufficiently "close" to a lattice (in the intuitive sense, formalised by
Chabauty topology In mathematics, the Chabauty topology is a certain topological structure introduced in 1950 by Claude Chabauty, on the set of all closed subgroups of a locally compact group ''G''. The intuitive idea may be seen in the case of the set of all latt ...
or by the topology on a
character variety In the mathematics of moduli theory, given an algebraic, reductive, Lie group G and a finitely generated group \pi, the G-''character variety of'' \pi is a space of equivalence classes of group homomorphisms from \pi to G: :\mathfrak(\pi,G)=\ ...
) is actually conjugated to the original lattice by an element of the ambient Lie group. A consequence of local rigidity and the Kazhdan-Margulis theorem is Wang's theorem: in a given group (with a fixed Haar measure), for any ''v>0'' there are only finitely many (up to conjugation) lattices with covolume bounded by ''v''. The
Mostow rigidity theorem Mostow may refer to: People * George Mostow (1923–2017), American mathematician ** Mostow rigidity theorem * Jonathan Mostow Jonathan Mostow (born November 28, 1961) is an American film director, screenwriter, and producer. He has directed f ...
states that for lattices in simple Lie groups not locally isomorphic to \mathrm_2(\mathbb R) (the group of 2 by 2 matrices with determinant 1) any isomorphism of lattices is essentially induced by an isomorphism between the groups themselves. In particular, a lattice in a Lie group "remembers" the ambient Lie group through its group structure. The first statement is sometimes called ''strong rigidity'' and is due to
George Mostow George Daniel Mostow (July 4, 1923 – April 4, 2017) was an American mathematician, renowned for his contributions to Lie theory. He was the Henry Ford II (emeritus) Professor of Mathematics at Yale University, a member of the National Academy o ...
and
Gopal Prasad Gopal Prasad (born 31 July 1945 in Ghazipur, India) is an Indian-American mathematician. His research interests span the fields of Lie groups, their discrete subgroups, algebraic groups, arithmetic groups, geometry of locally symmetric spaces ...
(Mostow proved it for cocompact lattices and Prasad extended it to the general case). ''
Superrigidity In mathematics, in the theory of discrete groups, superrigidity is a concept designed to show how a linear representation ρ of a discrete group Γ inside an algebraic group ''G'' can, under some circumstances, be as good as a representation of ''G' ...
'' provides (for Lie groups and algebraic groups over local fields of higher rank) a strengthening of both local and strong rigidity, dealing with arbitrary homomorphisms from a lattice in an algebraic group ''G'' into another algebraic group ''H''. It was proven by Grigori Margulis and is an essential ingredient in the proof of his arithmeticity theorem.


Nonrigidity in low dimensions

The only semisimple Lie groups for which Mostow rigidity does not hold are all groups locally isomorphic to \mathrm_2(\mathbb R). In this case there are in fact continuously many lattices and they give rise to
Teichmüller space In mathematics, the Teichmüller space T(S) of a (real) topological (or differential) surface S, is a space that parametrizes complex structures on S up to the action of homeomorphisms that are isotopic to the identity homeomorphism. Teichmülle ...
s. Nonuniform lattices in the group \mathrm_2(\mathbb C) are not locally rigid. In fact they are accumulation points (in the Chabauty topology) of lattices of smaller covolume, as demonstrated by hyperbolic Dehn surgery. As lattices in rank-one p-adic groups are virtually free groups they are very non-rigid.


Tree lattices


Definition

Let T be a tree with a cocompact group of automorphisms; for example, T can be a regular or biregular tree. The group of automorphisms\mathrm(T) of T is a locally compact group (when endowed with the
compact-open topology In mathematics, the compact-open topology is a topology defined on the set of continuous maps between two topological spaces. The compact-open topology is one of the commonly used topologies on function spaces, and is applied in homotopy theory and ...
, in which a basis of neighbourhoods of the identity is given by the stabilisers of finite subtrees, which are compact). Any group which is a lattice in some \mathrm(T) is then called a ''tree lattice''. The discreteness in this case is easy to see from the group action on the tree: a subgroup of \mathrm(T) is discrete if and only if all vertex stabilisers are finite groups. It is easily seen from the basic theory of group actions on trees that uniform tree lattices are virtually free groups. Thus the more interesting tree lattices are the non-uniform ones, equivalently those for which the quotient graph \Gamma \backslash T is infinite. The existence of such lattices is not easy to see.


Tree lattices from algebraic groups

If F is a local field of positive characteristic (i.e. a completion of a function field of a curve over a finite field, for example the field of formal
Laurent Laurent may refer to: *Laurent (name), a French masculine given name and a surname **Saint Laurence (aka: Saint ''Laurent''), the martyr Laurent **Pierre Alphonse Laurent, mathematician **Joseph Jean Pierre Laurent, amateur astronomer, discoverer ...
power series \mathbb F_p((t))) and G an algebraic group defined over F of F-split rank one, then any lattice in G is a tree lattice through its action on the
Bruhat–Tits building In mathematics, a building (also Tits building, named after Jacques Tits) is a combinatorial and geometric structure which simultaneously generalizes certain aspects of flag manifolds, finite projective planes, and Riemannian symmetric spaces. Bu ...
which in this case is a tree. In contrast to the characteristic 0 case such lattices can be nonuniform, and in this case they are never finitely generated.


Tree lattices from Bass–Serre theory

If \Gamma is the fundamental group of an infinite
graph of groups In geometric group theory, a graph of groups is an object consisting of a collection of groups indexed by the vertices and edges of a graph, together with a family of monomorphisms of the edge groups into the vertex groups. There is a unique group, ...
, all of whose vertex groups are finite, and under additional necessary assumptions on the index of the edge groups and the size of the vertex groups, then the action of \Gamma on the Bass-Serre tree associated to the graph of groups realises it as a tree lattice.


Existence criterion

More generally one can ask the following question: if H is a closed subgroup of \mathrm(T), under which conditions does H contain a lattice? The existence of a uniform lattice is equivalent to H being unimodular and the quotient H \backslash T being finite. The general existence theorem is more subtle: it is necessary and sufficient that H be unimodular, and that the quotient H \backslash T be of "finite volume" in a suitable sense (which can be expressed combinatorially in terms of the action of H), more general than the stronger condition that the quotient be finite (as proven by the very existence of nonuniform tree lattices).


Notes


References

* * * * * *{{cite book , last=Gelander , first=Tsachik , editor-last1=Bestvina , editor-first1=Mladen , editor-last2=Sageev , editor-first2=Michah , editor-last3=Vogtmann , editor-first3=Karen , title=Geometric group theory , chapter=Lectures on lattices and locally symmetric spaces , year=2014 , pages=249–282 , arxiv=1402.0962, bibcode=2014arXiv1402.0962G Algebraic groups Differential geometry Ergodic theory Geometric group theory Lie groups