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 $$INV(G)/FIN(G)$$