I am quite confused about this since I thought the Banach-Tarski theorem implies it is not but I also heard from multiple people that it is amenable since it is the semidirect product of an abelian and a compact group. It would be amazing if someone could explain the relationship between amenable groups and the paradox to me! I would also be interested in the amenability of $E(d)$ for general $d$.
2026-02-23 06:00:13.1771826413
Is the Euclidean group $E(3)$ on $\mathbb{R}^3$ amenable?
52 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail At
1
There are 1 best solutions below
Related Questions in GROUP-THEORY
- What is the intersection of the vertices of a face of a simplicial complex?
- Group with order $pq$ has subgroups of order $p$ and $q$
- How to construct a group whose "size" grows between polynomially and exponentially.
- Conjugacy class formula
- $G$ abelian when $Z(G)$ is a proper subset of $G$?
- A group of order 189 is not simple
- Minimal dimension needed for linearization of group action
- For a $G$ a finite subgroup of $\mathbb{GL}_2(\mathbb{R})$ of rank $3$, show that $f^2 = \textrm{Id}$ for all $f \in G$
- subgroups that contain a normal subgroup is also normal
- Could anyone give an **example** that a problem that can be solved by creating a new group?
Related Questions in AMENABILITY
- existence of weak* continuous means for amenable locally compact groups
- Polynomially sized Folner sets
- The positive net of a weak* convergent net is weak* convergent.
- Show that free group $\mathbb{F}_2$ is not amenable using Hulanicki-Reiter condition.
- HNN embedding theorem for countable groups not preserving amenability
- Amenability of countable product of a finite group.
- Bound on the measure of a sum of sets
- Extensions of amenable groups
- Strong Folner condition(SFC) implies the existence of a left Følner sequence.
- Index of subgroups in residually finite groups
Trending Questions
- Induction on the number of equations
- How to convince a math teacher of this simple and obvious fact?
- Find $E[XY|Y+Z=1 ]$
- Refuting the Anti-Cantor Cranks
- What are imaginary numbers?
- Determine the adjoint of $\tilde Q(x)$ for $\tilde Q(x)u:=(Qu)(x)$ where $Q:U→L^2(Ω,ℝ^d$ is a Hilbert-Schmidt operator and $U$ is a Hilbert space
- Why does this innovative method of subtraction from a third grader always work?
- How do we know that the number $1$ is not equal to the number $-1$?
- What are the Implications of having VΩ as a model for a theory?
- Defining a Galois Field based on primitive element versus polynomial?
- Can't find the relationship between two columns of numbers. Please Help
- Is computer science a branch of mathematics?
- Is there a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?
- Identification of a quadrilateral as a trapezoid, rectangle, or square
- Generator of inertia group in function field extension
Popular # Hahtags
second-order-logic
numerical-methods
puzzle
logic
probability
number-theory
winding-number
real-analysis
integration
calculus
complex-analysis
sequences-and-series
proof-writing
set-theory
functions
homotopy-theory
elementary-number-theory
ordinary-differential-equations
circles
derivatives
game-theory
definite-integrals
elementary-set-theory
limits
multivariable-calculus
geometry
algebraic-number-theory
proof-verification
partial-derivative
algebra-precalculus
Popular Questions
- What is the integral of 1/x?
- How many squares actually ARE in this picture? Is this a trick question with no right answer?
- Is a matrix multiplied with its transpose something special?
- What is the difference between independent and mutually exclusive events?
- Visually stunning math concepts which are easy to explain
- taylor series of $\ln(1+x)$?
- How to tell if a set of vectors spans a space?
- Calculus question taking derivative to find horizontal tangent line
- How to determine if a function is one-to-one?
- Determine if vectors are linearly independent
- What does it mean to have a determinant equal to zero?
- Is this Batman equation for real?
- How to find perpendicular vector to another vector?
- How to find mean and median from histogram
- How many sides does a circle have?
It is better to think about the group $SO(3)$. If you equip it with the natural matrix topology, it is compact, hence amenable: The invariant probability measure is the natural one, given by a left-invariant volume form on this Lie group. (There is a lot to digest here if you do not know what these words mean.) But as a discrete group, $SO(3)$ is not amenable since it contains a free group on two generators and the latter is nonamenable. The existence of nonabelian free subgroups is the heart of the proof of the Banach-Tarski paradox. Again, it requires quite a bit of work on your part to understand what am I talking about if you are not familiar with these notions.