User:Juan Marquez/ends

• Almost-invariant set
• set of ends
• number of ends
• Stalling theorem

The concept on end of a space is important because belongs to a serie of invariants called quasi-isometry

ends of a group

A subset's relation , with symbol $\scriptstyle \subset _{a}$, called almost-inclusion and almost-equality in the power set of a group is defined as

$\scriptstyle E\subset _{a}F$ means that $\scriptstyle E\cap F^{c}$ is a finite set

$\scriptstyle E=_{a}F\,$ means that $\scriptstyle E\subset _{a}F$ and $\scriptstyle F\subset _{a}E$

A subset $\scriptstyle E\in {\mathcal {P}}(G)$ is dubbed almost-invariant if and only if

for each $\scriptstyle g\in G$ it happens $Eg=_{a}E$

The set $\scriptstyle {\mathcal {P}}(G)$ together with the symmetric difference $\Delta$ is $\scriptstyle \mathbb {Z} _{2}$-vector space and it have the subspaces

$\scriptstyle INV(G)=\{E\in {\mathcal {P}}(G)|\ E$ is almost-invariant $\}\,$

$\scriptstyle FIN(G)=\{E\in {\mathcal {P}}(G)|\ E$ is finite $\}\,$

then the number of ends happens to be equals to the dimension of $INV(G)/FIN(G)$