** Geometry of nilpotent groups and groups of polynomial growth **

**1. Growth **

**Definition**. Let be a locally compact group equipped with a left Haar measure . Assume has a compact generating set . The growth function is .

Its asymptotic behaviour is independant on , where we write, for functions , if , and if .

The study of growth began in the 60’s with Milnor, Svarc, especially for discrete or connected Lie groups.

**Examples**. 1. has polynomial growth of degree .

2. The affine group of the line is a semi-direct product . It is the set of matrices where , . . The -ball contains , so growth is exponential.

3. The Heisenberg group is the set of matrices where , , . . The -ball contains , so growth is polynomial of degree 4.

**2. The case of Lie groups **

These 3 examples are representative of the general case for Lie groups.

Theorem 1 (Guivarc’h, 1973, then a PhD student in Rennes)Let be a virtually connected Lie group. Then has polynomial growth if and only if all eigenvalues of the adjoint representation of on belong to the unit circle. Otherwise, has exponential growth.

**Remark**. Nilpotent Lie groups correspond to the case where 1 is the only eigenvalue. The isometry group of is an example where eigenvalues can be non real.

**3. Gromov’s polynomial growth theorem **

Theorem 2 (Gromov 1981)Let be a discrete, finitely generated group with polynomial growth. Then is virtually nilpotent (and conversely).

**Question**. Does nonpolynomial growth imply exponential growth ?

Theorem 3 (Tits 1972, Grigorchuk 1984)Answer is yes for linear groups.Answer is no for general discrete groups.

Open for finitely presented groups.

**4. Generalization to locally compact groups **

Theorem 4 (Losert 1987)Let be a locally compact, compactly generated group with polynomial growth. Then is compact-by-Lie, i.e. is has a compact normal subgroup such that is Lie. (Here Lie, means Lie, i.e. connected Lie-by-discrete.)Let be a compactly generated Lie group. Then has polynomial growth if and only if

- has polynomial growth,
- the discrete quotient is virually nilpotent.
- the action of on has its eigenvalues on the unit circle.

**5. Malcev completion **

Theorem 5Let be a finitely generated nilpotent group. Then there is a homomorphism where

- has finite kernel (the set of torsion elements of ),
- is isomorphic to the set of rational points of an unipotent algebraic group,
- is nilpotent, uniquely divisible ( is injective for all ), the image intersects every nontrivial subgroup of ,
- is a cocompact lattice in ,
- the sequence is functorial.

Definition 6Say that a locally compact group of polynomial growth is universal if it is a semidirect product where is a simply connected nilpotent Lie group and is a maximal compact group of automorphisms of . is called the nilshadow of .

Proposition 7 (Essentially due to Breuillard)Every locally compact group of polynomial growth admits an essentially unique homomorphism to a universal one, with compact kernel and cocompact image.