Exercise 37, § 7

Let $G$ be a group and $S$ a finite symmetric subset of $G$ generating $G$. Let $\mathfrak{L}$ denote the set of finite subsets of $G$, ordered by inclusion; it is a directed set.

(a) Let $T \in \mathfrak{L}$ and let $B_T$ be the set of connected $S$-components of $G - T$ (cf. Exercise 18); let $B_T^\infty = B_T - (\mathfrak{L} \cap B_T)$. Show that $B_T$ is finite (note that, if $T \neq \varnothing$ and $Z \in B_T$, there exists $t \in T$ and $z \in Z$ which are $S$-neighbours).

(b) Let $T, T' \in \mathfrak{L}$ with $T \subset T'$. If $Z' \in B_{T'}$, show 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 $B_T^\infty$. The inverse limit of the $B_T$, with $T$ running through $\mathfrak{L}$, is denoted by $B$. Show 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 non-empty if and only if $G$ is infinite.

(c) Let $T$ be a non-empty $S$-connected finite subset of $G$ and let $Z \in B_T^\infty$. Show that there exist $g \in G$ such that $gT \cap T = \varnothing$ and $gT \cap Z \neq \varnothing$ and that then $gT \subset Z$. Show 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 $Z_1$, then $gZ' \subset Z$. Show that, writing $T' = T \cup gT$, 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 symmetric finite subset of $G$ generating $G$ and $B'$ the set of ends of $G$ relative to $S'$. Define a bijection of $B$ onto $B'$ (reduce it to the case where $S \subset S'$). The cardinal of $B$ is thus independent of the choice of $S$; it is called the number of ends of $G$.

(e) Show, using (c), that the number of ends of $G$ is equal to 0, 1, 2 or $2^{k_0}$.

(f) Show that the number of ends of $\mathbf{Z}$ is 2 and that that of $\mathbf{Z}^n$ ($n \geq 2$) is 1.