(following an outline by Wouter van Limbeek; filling this outline is a work-in-progress. Sections 6 and 7 are particularly incomplete; sections 11 and 12—or 8 onwards—could [should?] be split to form its own post.)

Here we are working with connected Lie groups.

**1) Classes of Lie groups: abelian, nilpotent, solvable, semisimple**

Abelian Lie groups are completely classified: these all have the form (where denotes the *e*-dimensional torus.)

**Nilpotent** groups are those with a terminating lower central series . Examples of nilpotent Lie groups: Heisenberg groups, unitriangular groups, the (3-by-3) Heisenberg group mod its center.

**Solvable** groups are those with a terminating upper central series . There is an equivalent formulation in terms of composition series—in this sense solvable groups are built up from (simple) abelian groups. Examples of solvable Lie groups:

- , where acts on as the matrix .
- , where the action is by scalar multiplication.
- The group of all upper-triangular matrices.
- , lifted from , where the action is by rotation.

By definition, we have abelian nilpotent solvable

**Simplicity and semisimplicity**

A Lie algebra is **simple** if it has no proper ideals (i.e. proper subspaces closed under the Lie bracket.) is **semisimple** if it is a direct sum of simple Lie algebras.

A Lie group *G* is (semi)simple if its Lie algebra is (semi)simple.

Equivalently, *G* is simple if it has no nontrivial **connected** normal subgroups (what can go wrong if we don’t have that additional adjective?)

*G* is semisimple if its universal cover is a direct product of simple Lie groups. (Note that things can go wrong if we do not take the universal cover, e.g. the quotient of by , where the action is the diagonalisation of , is semisimple [in the sense that its Lie algebra is semisimple], but not a product.)

**Relation between these types and the adjoint representation.**

Recall the adjoint representation given by sending a group element *g* to the matrix representing , where denotes conjugation by *g*. Now

*G* is abelian iff Ad(*G*) is the trivial representation
*G* is nilpotent iff Ad(*G*) is unipotent
*G* is solvable iff Ad(*G*) is (simultaneously) triangularizable (over an algebraically-closed field)
*G* is semisimple iff Ad(*G*) is a semisimple representation (i.e. may be written as a direct sum of irreducible Ad-invariant representations.)
*G* contains a lattice (see below) iff Ad(*G*) .

**2) Levi decomposition of Lie groups**

Note that subgroups of solvable groups are solvable, and extensions of solvable groups by solvable groups are solvable. Putting all of these together, we obtain that any Lie group *G* has a unique maximal closed connected normal solvable subgroup *R* (called the **solvable radical** of *G*.)

The **Levi** **decomposition **(also called, in some contexts, the Levi-Malcev decomposition) of *G* is *G = RS*, where *R* is the solvable radical, is semisimple, and is discrete.

If G is simply-connected, then .

**3) Classification of compact Lie groups**

Any compact connected solvable Lie group G is a torus (Proof sketch: Induct on length of upper central series. For *n=*1, *G* is abelian and we are done. Otherwise, we have the short exact sequence ; the outer terms are tori and , and we have a map . Since this is a group homomorphism from a connected group to a discrete group, it musth ave trivial image; hence is central, and we can induce on to produce a section .

Compact connected semisimple Lie groups can also be classified, but this story involves much more algebra (representation theory, highest weights, etc.

Using the Levi decomposition, we have the following

**Theorem**. Any compact connected Lie group *G* is isomorphic to , where *F* is a finite group and are simple.

**4) When is a Lie group an algebraic group? When is a Lie group linear?**

An **algebraic group** is a group with a compatible algebraic variety structure (i.e. multiplication and inversion are regular maps.) A **linear group** is (isomorphic to) a subgroup of GL(*n,k*) for some field *k*. Note GL(*n,k*) is algebraic, and thus linear groups defined by polynomial equations (but not all linear groups!) are algebraic.

A Lie group is algebraic if it is isomorphic to a linear algebraic group (but this is not an iff [?])

Many of the classical Lie groups are (linear) algebraic groups, but note that not all Lie groups are algebraic: e.g. is not algebraic because its center is not algebraic (“is too large to be algebraic.”)

**5) When is the exponential map a diffeomorphism?**

We have an exponential map . Its derivative at the identity, , is the identity map; exp therefore restricts to a diffeomorphism from some neighborhood of 0 in to a neighborhood of 1 in *G*.

If *G* is connected, simply-connected, and nilpotent, the exponential map exp is a (global) diffeomorphism (in fact, an analytic isomorphism of analytic manifolds, if *G* is linear algebraic.)

**6) Topology of Lie groups: fundamental group, homotopy type, cohomology … **

**Theorem**. Any connected Lie group has abelian fundamental group.

**Proof sketch**:** **In fact this is true for any connected topological group, because the group structure forces things to be nice that way. To make this precise can be mildly annoying though.

**Theorem (Weyl).** The fundamental group of a compact semisimple Lie group is finite

**Theorem.** Any connected Lie group has trivial second homotopy and torsionfree third homotopy.

**Proof sketch: **Since any Lie group retracts onto its maximal compact subgroup, WLOG we work with only connected groups.

From the long exact sequence of homotopy groups obtained from the path-loop fibration, . By Morse theory, and .

Many more things can be said: see e.g. this survey.

**Relation with the Lie algebra**

**7) Geometry of Lie groups: relation between geometry of an invariant metric and algebraic structure.**

Note that a Lie group structure yields a natural left-invariant (or right-invariant) metric, given by propagating the inner product at the identity by group multiplication. With additional hypotheses, we can make this bi-invariant:

**Theorem. **Every compact Lie group *G* admits a bi-invariant metric.

**Idea of proof** Use the natural Haar measure on *G* to average the left-invariant metric.

We can play this invariant metric and the group structure off against each other: e.g. the geodesics of *G* coincide with the left-translates of 1-parameter subgroups of *G *(at the identity); as a corollary of this, we obtain that any Lie group *G* is (geodesically) complete.

**8) Selberg’s lemma**

**Lemma (Selberg 1960).** A finitely generated linear group over a field of zero characteristic is virtually torsion-free.

(cf. **Theorem (Malcev 1940).** A finitely generated linear group is residually finite.)

**Proof:** Using number fields (local fields, see Cassels or Ratcliffe), or Platonov’s theorem, which seems to be a bunch of commutative algebra (see Bogdan Nica’s paper.)

**9) Finite generation/presentation of lattices: Milnor-Svarc and Borel-Serre.**

A **lattice** is a discrete group with of finite volume (as measured by the natural Haar measure on G.) The prototypical example is , which of course is the isometry group of the torus.

**Facts**

- Any lattice in a solvable group is co-compact (also called uniform.)
- (Borel, Harish-Chandra) Any noncompact semisimple group contains a lattice that is not co-compact (also called non-uniform.)

**Theorem (Milnor-Svarc).** If acts on a proper geodesic metric space X properly discontinuously and cocompactly, then is finitely-generated.

**Proof: **Since cocompactly, the action has a compact fundamental domain . By the proper discontinuity of the action, the vertices of Z cannot accumulate. But compactness then implies Z has finitely many vertices, and hence finitely many sides. Since each algebraically independent generator would add a side to Z, this implies that is finitely generated.

In fact, many non-uniform lattices are also finitely-generated.

The **-rank **of a Lie group *G *is the dimension of the largest abelian subgroup of *Z(a)* simultaneously diagonalizable over , as *a* varies over all semisimple elements of *G* ( is semisimple if Ad(*a*) is diagonalizable over . Geometrically, we may interpret the -rank as the dimension of the largest flat in *G*.

A locally-compact group *G* is said to have property (T) if for every continuous unitary representation into some Hilbert space there exists and compact s.t. if with s.t. for all , then with s.t. fixes *v*‘. (i.e., the existence of an almost-invariant vector implies the existence of a fixed point.)

**Theorem (Kazhdan, 1968):** If *G* is a simple Lie group of -rank , and is a lattice, then has property (T), and hence is finitely-generated.

Alternative proof: every such lattice is arithmetic, then use a fundamental domain and argue [as before]

**10) Levi-Malcev decomposition of lattices**

The Levi-Malcev decomposition as applied to a lattice tells us that where is a lattice in a solvable group and is a lattice in a semisimple group: note that each of these intersections remains discrete, and has finite co-volume for if not would not have finite co-volume.

Hence the general scheme for understanding lattices in Lie groups: first understand lattices in solvable groups and in semisimple groups, then piece them together using the Levi-Malcev decomposition …

**11) (Non-)arithmetic lattices. When is a lattice arithmetic?**

Most generally, an arithmetic subgroup of a linear algebraic group *G* defined over a number field *K* is a subgroup *Γ* of *G(K)* that is commensurable with , where is the ring of integers of *K*.

Hence we may define more abstractly a lattice in a connected (solvable) Lie group *G *is said to be **arithmetic** if there exists a cocompact faithful representation *i* of *G* into (where is an algebraic subgroup) with closed image and s.t. has finite index in both and .

We have that

**Theorem (Borel, Harish-Chandra).** Arithmetic subgroups are lattices.

Conversely,

**Theorem (Mostow). ***(4.34 in Raghunathan.)*** **Let *G* be a simply-connected solvable Lie group and a lattice. Then *G* admits a faithful representation into which sends into .

Note, however, the counterexample on pp. 76-77 of Raghunathan.

Much stronger results are true in semisimple Lie groups:

**Theorem (Margulis arithmeticity). **Any irreducible lattice in a semisimple Lie group with no rank one factors is arithmetic.

(More precisely, see statement of Selberg’s conjectures in Section 7 of this article.)

**Theorem (Margulis’ commensurator criterion.) **Let be an irreducible lattice (where *G* may have rank 1). Then is arithmetic iff the commensurator of is dense in G.

Recall the commensurator of is the subset of *G* consisting of elements *g* s.t. and are commensurable

**12) Rigidity of lattices**

(i.e. when can you deform a lattice in the ambient Lie group?)

**Theorem (Weil local rigidity).** Let be a finitely generated group, *G* a Lie group and be a homomorphism. Then is locally rigid if . Here is the Lie algebra of *G* and acts on by .

**Theorem (Mostow rigidity).** Let Γ and Δ be discrete subgroups of the isometry group of (with *n* > 2) whose quotients and have finite volume. If Γ and Δ are isomorphic as discrete groups, then they are conjugate.

i.e. lattices in hyperbolic space are pretty darn rigid. But then there’s even more. A lattice is caled **irreducible** if no finite index subgroup is a product;

**Theorem (Margulis superrigidity).** Let Γ be an irreducible lattice in a connected semisimple Lie group *G* of rank at least 2, with trivial center, and without compact factors. Suppose *k* is a local field.

Then any homomorphism π of Γ into a noncompact *k*-simple group over *k* with Zariski dense image either has precompact image, or extends to a homomorphism of the ambient groups.

**Remark. **Margulis superrigidity implies Margulis arithmeticity (how?)

**Margulis’ normal subgroup theorem**. Let G be a connected semisimple Lie group of rank > 1 with finite center, and let be an irreducible lattice.

If is a normal subgroup of , then either N lies in the center of G (and hence is finite), or the quotient is finite.

Or, in short: “many lattices in semisimple Lie groups are simple, up to finite error.”

Some comments here

**Quasi-isometric rigidty of lattices**

See survey by Benson Farb

**Zariski tangent space to representation variety and cohomology.**

**References**