Intereting Posts

The Multiplier Algebra of the Hardy Hilbert Space
system of congruences proof
$\pi$ in arbitrary metric spaces
Antiderivative of $e^{x^2}$: Correct or fallacy?
In a finite ring extension there are only finitely many prime ideals lying over a given prime ideal
prove the inverse image of a maximal ideal is also a maximal ideal
Space of continuous functions with compact support dense in space of continuous functions vanishing at infinity
counterexample to RH; how big would it have to be?
Homology of connected sums
proof of the Meyers-Serrin Theorem in Evans's PDE book
Show $\nabla^2g=-f$ almost everywhere
Is there finest topology which makes given vector space into a topological vector space?
All finite abelian groups of order 1024
Monos in $\mathsf{Mon}$ are injective homomorphisms.
Global sections of $\mathcal{O}(-1)$ and $\mathcal{O}(1)$, understanding structure sheaves and twisting.

Let $G$ be a finitely generated amenable group.

I know that it’s a basic result that every quotient of $G$ is amenable.

Is it also true that every Folner sequence of $G$ projects onto a Folner sequence of the quotient? Is there a direct proof of that?

- Can RUBIK's cube be solved using group theory?
- Why should I consider the components $j^2$ and $k^2$ to be $=-1$ in the search for quaternions?
- Two finite abelian groups with the same number of elements of any order are isomorphic
- Generating function for permutations in $S_n$ with $k$ cycles.
- Number of elements which are cubes/higher powers in a finite field.
- Necessary and Sufficient Condition for $\phi(i) = g^i$ as a homomorphism - Fraleigh p. 135 13.55
- Example in which a normal subgroup acts non-equivalent on its orbits
- Abelian subgroups of $GL_n(\mathbb{F}_p)$
- Right identity and Right inverse implies a group
- What does the plus sign contained in a circle ($\oplus$) mean in this case?

It behaves quite bad. For instance, consider the projection $\mathbf{Z}^2\to\mathbf{Z}$, $(m,n)\mapsto n$, and $F_n=\{1,\dots,n\}^2\cup (J_n\times\{0\})$, where $J_n$ is a subset of the set of even negative integers, of cardinal $\ll n^2$. Then $(F_n)$ is a Følner sequence but its projection is not.

It’s just that this is not the good point of view. The closest point of view is replacing almost invariant subsets with almost invariant probabilities (a probability on a discrete is just a non-negative $\ell^1$-function with sum 1), and the latter behaves well with push-forwards because it takes into account the cardinal of the fibers. In the previous case, if we pass from the Følner subsets to the associated uniform probabilities, (by normalizing the indicator function of $F_n$), the push-forward is no longer an indicator function and gives negligible weight to the parasite subset $J_n$.

In general, as above, passing from Følner subsets to almost invariant probabilities is trivial; while the converse is more involved but now classical.

- another balls and bins question
- Prove that $1^2 + 2^2 + … + (n-1)^2 < \frac {n^3} { 3} < 1^2 + 2^2 + … + n^2$
- Dimension of set of commutable matrices
- Can any linear transformation be represented by a matrix?
- Evaluate limit of $(2\sin x\log \cos x + x^{3})/x^{7}$ as $x \to 0$
- Is $x^{1-\frac{1}{n}}+ (1-x)^{1-\frac{1}{n}}$ always irrational if $x$ is rational?
- How does one combine proportionality?
- Probability puzzle – the 3 cannons
- What's so special about primes $x^2+27y^2 = 31,43, 109, 157,\dots$ for cubics?
- Algorithm to compute Gamma function
- What is the probability that the resulting four line segments are the sides of a quadrilateral?
- Inverse Trigonometric Integrals
- Hanging a picture on the wall using two nails in such a way that removing any nail makes the picture fall down
- On the greatest norm element of weakly compact set
- Why this two surfaces have one end?