= Solution
The <exchange condition for a Coxeter group> says that if $w=s_1\cdots s_n$ is reduced and $s$ is simple with $\ell(ws)<\ell(w)$, then
$$
ws=s_1\cdots\widehat{s_j}\cdots s_n
$$
for some $j$. The <Matsumoto theorem> says that any two reduced expressions for the same element are connected by <braid relation in a Coxeter group>[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 word
$$
u_N=(s\,a\,b\,s\,c\,d)^N.
$$
Between 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.
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