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

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