User:Juan Marquez/ends

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 ${\displaystyle \scriptstyle \subset _{a}}$, called almost-inclusion and almost-equality in the power set of a group is defined as

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

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

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

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

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

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