Locally parabolic subgroups in Coxeter groups of arbitrary ranks

Despite the significance of the notion of parabolic closures in Coxeter groups of finite ranks, the parabolic closure is not guaranteed to exist as a parabolic subgroup in a general case. In this paper, first we give a concrete example to clarify that the parabolic closure of even an irreducible reflection subgroup of countable rank does not necessarily exist as a parabolic subgroup. Then we propose a generalized notion of "locally parabolic closure" by introducing a notion of "locally parabolic subgroups", which involves parabolic ones as a special case, and prove that the locally parabolic closure always exists as a locally parabolic subgroup. It is a subgroup of parabolic closure, and we give another example to show that the inclusion may be strict in general. Our result suggests that locally parabolic closure has more natural properties and provides more information than parabolic closure. We also give a result on maximal locally finite, locally parabolic subgroups in Coxeter groups, which generalizes a similar well-known fact on maximal finite parabolic subgroups.