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Coxeter groups are a very special class of groups defined via generators and relations. They are intimately related to the classical finite dimensional semisimple Lie algebras and to their infinite dimensional counterparts - the affine and indefinite Kac-Moody algebras. In this talk, we focus on questions related to the "growth type" of such groups; roughly, if gamma(n) denotes the number of elements of the group of length <=n, the growth type measures the rate of growth of gamma(n) with n.

It is well known that Coxeter groups of finite and affine type have polynomial growth while all other Coxeter groups have exponential growth. I will describe a generalization of this latter result to quotients $W/W_J$ of such Coxeter groups by their parabolic subgroups. Along the way, we will introduce reflection subgroups of $W$ and a criterion of M. Dyer in terms of inner products of roots that enables construction of such subgroups. We'll use this and some root system combinatorics to construct certain reflection subgroups of $W$ that are isomorphic to "universal Coxeter groups". This will lead us to the result that the quotients $W/W_J$ have exponential growth too.