Geometry of nilpotent groups and groups of polynomial 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 5 Let 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 6 Say 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.