Notes of Yves Cornulier’s Rennes lecture nr 1

Geometry of nilpotent groups and groups of polynomial growth

1. Growth

Definition. Let {G} be a locally compact group equipped with a left Haar measure {\lambda}. Assume {G} has a compact generating set {S}. The growth function is {b_S(n)=\lambda(S^n)}.

Its asymptotic behaviour is independant on {S}, where we write, for functions {{\mathbb R}\rightarrow{\mathbb R}}, {f\leq g} if {f(n)\leq Cg(C'n)+C''}, and {f\sim g} if {f\leq g\leq f}.

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

Examples. 1. {{\mathbb R}^d} has polynomial growth of degree {d}.

2. The affine group of the line is a semi-direct product {{\mathbb R}\times{\mathbb R}}. It is the set of matrices {\begin{pmatrix} 1 & x \\ 0 & e^t \end{pmatrix}} where {x}, {t\in{\mathbb R}}. {\lambda=dx\,dt}. The {n}-ball contains {\{(x,t)\,;\,-e^n \leq x\leq e^n,\,-n\leq t\leq n\}}, so growth is exponential.

3. The Heisenberg group is the set of matrices {\begin{pmatrix} 1 & x &z\\ 0 & 1&y\\ 0&0&1 \end{pmatrix}} where {x}, {y}, {z\in{\mathbb R}}. {\lambda=dx\,dy\,dz}. The {n}-ball contains {\{(x,y,z)\,;\,|x|\leq n,\,|y|\leq n,\,|z|\leq n^2,\}}, 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 {G} be a virtually connected Lie group. Then {G} has polynomial growth if and only if all eigenvalues of the adjoint representation of {G} on {\mathfrak{g}} belong to the unit circle. Otherwise, {G} has exponential growth.

Remark. Nilpotent Lie groups correspond to the case where 1 is the only eigenvalue. The isometry group of {{\mathbb R}^d} is an example where eigenvalues can be non real.

3. Gromov’s polynomial growth theorem

Theorem 2 (Gromov 1981) Let {G} be a discrete, finitely generated group with polynomial growth. Then {G} 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 {G} be a locally compact, compactly generated group with polynomial growth. Then {G} is compact-by-Lie, i.e. is has a compact normal subgroup {W} such that {G/W} is Lie. (Here Lie, means Lie, i.e. connected Lie-by-discrete.)

Let {G} be a compactly generated Lie group. Then {G} has polynomial growth if and only if

  1. {G^0} has polynomial growth,
  2. the discrete quotient {G/G^0} is virually nilpotent.
  3. the action of {G} on {\mathfrak{g}} has its eigenvalues on the unit circle.


5. Malcev completion

Theorem 5 Let {\Gamma} be a finitely generated nilpotent group. Then there is a homomorphism {\Gamma\rightarrow\Gamma_{\mathbb Q}\rightarrow\Gamma_{\mathbb R}} where

  1. {i:\Gamma\rightarrow\Gamma_{\mathbb Q}} has finite kernel (the set of torsion elements of {\Gamma}),
  2. {\Gamma_{\mathbb Q}} is isomorphic to the set of rational points of an unipotent algebraic group,
  3. {\Gamma_{\mathbb Q}} is nilpotent, uniquely divisible ({x\mapsto x^n} is injective for all {n}), the image {i(\Gamma)} intersects every nontrivial subgroup of {\Gamma_{\mathbb Q}},
  4. {i(\Gamma)} is a cocompact lattice in {\Gamma_{\mathbb R}},
  5. the sequence is functorial.


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

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


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