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The deletion condition for involutory generators says that whenever a wordis not reduced, there are indices $i<j$ such that deleting $s_i$ and $s_j$ leaves a word representing the same group element.

The exchange condition for a Coxeter group says that if $w=s_1\cdots s_m$ is reduced and $s\in S$ satisfies ${\ell }_S(sw)<{\ell }_S(w)$, thenfor some $i$. There is an equivalent right-handed form for ${\ell }_S(ws)<{\ell }_S(w)$.

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

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.