DYNAMICS OF GROUP ACTIONS – LECTURE 1 SUMMARY
1. Basic terminology and set-up
Definition 1.1. We say that a group G acts on a set X, notation G y X, if for every
g ∈ G there is a bijection Tg : X → X, such that Te = Id, where e ∈ G is the group unit
and Id is the identity map, and
Tg1 g2 = Tg1 ◦ Tg2 for all g1 , g2 ∈ G.
More formally we can say that there is a homomorphism from G into the group of bijections
of X. We usually omit the symbol T in notation and simply right g ·x instead of Tg (x),
gA instead of Tg (A), etc.
Given G y X, the set OG (x) = {g·x : g ∈ G} is called the orbit of x under the action.
We will consider metric groups, that is, assume that G is equipped with a metric, making
the group operations continuous.
Definition 1.2. Let X be a σ-compact metric space (i.e., a countable union of compact
sets) and G a metric group. We say that G y X is a continuous action if the map
G × X → X defined by
(g, x) 7→ g·x
is continuous.
Examples. (i) Every group acts on itself G y G by left and right multiplication:
Lg : x 7→ gx;
Rg : x 7→ xg −1 ,
x ∈ G.
(Check that this is the correct way to fit with the definition of the action!)
(ii) Standard notation: GLn (R) (or GL(n, R)) is the group of n × n invertible matrices
with real entries; it acts on the vector space Rn by multiplication (A, x) 7→ Ax.
Other classical groups of matrices: SLn (R) — matrices with determinant one, SOn (R)
— orthogonal matrices (n-dimensional rotations); they also act on Rn ; the group SOn (R)
acts on the unit sphere S n−1 in Rn . We also have groups of matrices with integer entries,
or complex entries, denoted GLn (Z), GLn (C), etc.
Date: March 23, 2017.
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DYNAMICS OF GROUP ACTIONS – LECTURE 1 SUMMARY
E(n) is group of isometries of Rn , which is generated by translations, rotations and
reflections.
(iii) Let X be a metric space. Consider the group of homeomorphisms of X (ignore
topology for now), denoted Homeo(X), which acts on X naturally.
For a finite set
T1 , . . . , Tm ∈ Homeo(X) denote by hT1 , . . . , Tm i the subgroup of Homeo(X) generated
by these maps. This is a countable group, which we consider with the discrete metric.
Definition 1.3. Let X be a σ-compact metric space. Denote by M(X) the set of Borel
probability measures on X. If we have G y X, a measure µ ∈ M(X) is called invariant
under G if g∗ µ = µ for all g ∈ G, where (g∗ µ) = µ(g −1 ·A) for any Borel set A ⊆ X. We
write MG (X) for the set of G-invariant measures.
It turns out that invariant measures do not necessarily exist, even for a continuous action
G y X on a compact metric space (see Exercise 1). In fact, one of the characterizations
of an amenable group, as we will see later, is that MG (X) is nonempty for a compact
space X.
2. Banach-Tarski paradox
In 1924 Banach and Tarski proved a theorem which states that “given a ball in R3 , it
can be decomposed into finitely many disjoint pieces that can be rearranged by Euclidean
isometries to form two balls of the same size as the original one.” This shows that there
is no finitely additive measure on R3 defined on P (X) (the collection of all subsets of X),
which is invariant under isometries and gives the unit ball a non-zero measure.
This is again related to the notion of amenability: E(3) is non-amenable, whereas
E(2), the group of planar isometries, is amenable, and there is no such “paradoxical
decomposition” in the plane.
See [1] and [2, Chapter 3] for the proof of Banach-Tarski paradox.
References
[1] Alejandra Garrido, An introduction to amenable groups, Preprint, 2013, available at
http://www.http://people.maths.ox.ac.uk/kar/amenable.pdf
[2] Kate Juschenko, A book in amenability (in preparation), available at
http://www.math.northwestern.edu/~juschenk/book.html