The following theorem was proved independently by [Deodhar][1] and by [Dyer][2]:

> Let $(W, S)$ be a Coxeter group. Let $V$ be a subgroup of $W$ generated by reflections. 
> Then there is a set $R$ of reflections in $V$ so that $(V, R)$ is a Coxeter group. 
> $R$ can be characterized uniquely as follows: Let $T$ be the set of reflections in $W$ 
> and, for $w \in W$, let $in(w)$ be the set of inversions of $w$.
> An element $t \in T$ lies in $R$ if and only if $in(t) \cap V = \{ t \}$.

Even when $S$ is finite, $R$ can be infinite! If I haven't messed up, one example is to take $W$ to be the Coxeter group on generators $p$, $q$, $r$, with no relations other than $p^2=q^2=r^2=e$. Let $V$ be the subgroup generated by
$$\cdots qpqprpqpq, qpqrqpq, qprpq, qrq, r, prp, pqrqp, pqprpqp, pqpqrqpqp, \cdots.$$
If I haven't messed up, $V$ is an infinite rank subgroup of $W$, with the set $R$ being the list of generators above.


Since these two papers have 86 citations in Mathscinet between them, I'd say this result is useful.

  [1]: http://www.ams.org/mathscinet-getitem?mr=1023969
  [2]: http://www.ams.org/mathscinet-getitem?mr=1076077