Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 111 / 2

The Cayley graph ${\mathrm{Cay}}_S(W)$ has vertex set $W$ and an unoriented edge $\{w,ws\}$ for every $w\in W$ and $s\in S$.

For a reflection of a Coxeter group $r\in R$, its wall isEquivalently, these are the edges fixed setwise and reversed by left multiplication by $r$. Removing their interiors separates the Cayley graph into two half-spaces.

Let $w=s_1\cdots s_m$ be reduced and follow the corresponding geodesic from $e$ to $w$. Its $i$th edge belongs to the wall ofA reduced path crosses each wall at most once, and these $m$ walls are exactly the walls separating its endpoints. Moreoverso $\ell (r_{i}w)<\ell (w)$. Conversely, if $\ell (rw)<\ell (w)$, write $r=us{u}^{-1}$ and apply the exchange condition for a Coxeter group to a reduced expression: it identifies $r$ with one of the reflections $r_i$. Therefore

By part (b), the hypothesis says that every wall separates $e$ from $w_0$. Let $u\in W$. For each wall, $e$ and $w_0$ lie in opposite half-spaces, so $u$ lies on exactly one of their two sides. The wall therefore separates exactly one of the pairs $(e,u)$ and $(u,w_0)$. Since Coxeter length equals the number of separating walls,