Exercise 37, § 7

Let $G$ be a group and $S$ a finite symmetric subset of $G$, generating $G$. We denote by $\mathfrak{S}$ the set of finite subsets of $G$, ordered by inclusion; it is a filtered set. a) Let $T \in \mathfrak{S}$, and let $B_T$ be the set of $S$-connected components of $G - T$ (cf. I, p. 146, Exercise 18); let $B_T^\infty = B_T - (\mathfrak{S} \cap B_T)$. Prove that $B_T$ is finite (notice that, if $T \neq \emptyset$ and if $Z \in B_T$, there exist $t \in T$ and $z \in Z$ which are $S$-neighbours). b) Let $T, T'$ be in $\mathfrak{S}$, such that $T \subset T'$. If $Z' \in B_{T'}$, now prove that there exists a unique element $Z \in B_T$ such that $Z' \subset Z$; the mapping $f_{TT'} : B_{T'} \to B_T$ such that $f_{TT'}(Z') = Z$ maps $B_{T'}^\infty$ onto $B_T^\infty$. We denote by $B$ the projective limit of the $B_T$, for $T$ ranging over $\mathfrak{S}$. Now prove that $B = \lim \leftarrow B_T^\infty$ and that the image of $B$ in $B_T$ is $B_T^\infty$. The set $B$ is called the set of ends of $G$; it is nonempty if and only if $G$ is infinite. c) Let $T$ be a finite, $S$-connected and nonempty subset of $G$, and let $Z \in B_T^\infty$. Now prove that there exists $g \in G$ such that $gT \cap T = \emptyset$ and $gT \cap Z \neq \emptyset$ and that one then has $gT \subset Z$. Prove that there exists $Z_1 \in B_T$ such that $gZ_1$ contains $T$ and all the $Z' \in B_T$ such that $Z' \neq Z$ (observe that $T \cup \bigcup_{Z' \neq Z} Z'$ is $S$-connected and does not meet $gT$); deduce that, if $Z' \in B_T$ is different from $Z_1$, one has $gZ' \subset Z$. Now prove that, if one puts $T' = T \cup gT$, the inverse image of $Z$ in $B_{T'}^\infty$ consists of at least $n-1$ elements, where $n = \mathrm{Card}(B_T^\infty)$. d) Let $S'$ be a finite symmetric subset of $G$ generating $G$ and let $B'$ be the set of ends of $G$ relative to $S'$. Define a bijection of $B$ onto $B'$ (reduce to the case where $S \subset S'$). The cardinal of $B$ is therefore independent of the choice of $S$; it is called the number of ends of $G$. e) Prove, using c), that the number of ends of $G$ is equal to $0, 1, 2$ or $2^{N_0}$. f) Prove that the number of ends of $\mathbf{Z}$ is 2 and that of $\mathbf{Z}^n$ ($n \geq 2$) is 1.