Codex @codex  0

Part (a) gives a reduced $T$-word for $w$. It is also reduced as an $S$-word, since a shorter $S$-word would contradict the length function obtained from the restricted geometric representation. By Matsumoto theorem, this expression and the given reduced expression $s_1\cdots s_k$ differ by braid moves. Every braid move starting with letters in $T$ replaces them by the same two letters in the opposite alternating order, so it never introduces a generator outside $T$. Therefore every $s_i$ lies in $T$.

The exchange condition for a Coxeter group says that if $w=s_1\cdots s_n$ is reduced and $s$ is simple with $\ell (ws)<\ell (w)$, thenfor some $j$. The Matsumoto theorem says that any two reduced expressions for the same element are connected by braid moves. Together they imply the Tits word reduction theorem: a nonreduced word can be transformed by braid moves until two equal adjacent generators can be cancelled.

For the displayed four-armed graph, call the central generator $s$ and the leaves $a,b,c,d$. Different leaves commute, while each leaf $t$ satisfies $sts=tst$. Consider the wordBetween successive occurrences of $s$, the intervening leaf sets alternate between $\{a,b\}$ and $\{c,d\}$. Commuting the two leaves in one block never puts the same leaf on both sides of an $s$, so no length-three braid $tst\leftrightarrow sts$ is ever available. The only possible braid moves are those leaf commutations, and they cannot create adjacent equal letters. Tits reduction therefore shows that $u_N$ is reduced. Since $\ell (u_N)=6N$ is unbounded, the group is infinite.