  We study extended Weyl groups of domestic and wild type. We start with an
extended Coxeter-Dynkin diagram, attach to it a set of roots (vectors) in an
$\mathbb{R}$-space and define the extended Weyl groups as groups that are
generated by the reflections related to these roots, the so called simple
reflections. We relate to such a group $W$ a generalized root system, and
determine the structure of $W$. In particular we present a normal form for the
elements of $W$. Further we define Coxeter transformations in $W$, and show the
transitivity of the Hurwitz action on the set of reduced reflection
factorizations of a Coxeter transformation where the reflections are the
conjugates of the simple reflections in $W$.
  We give an application of our results in the representation theory of
hereditary categories by establishing a bijection between the poset of certain
thick subcategories of a hereditary category, namely the category of coherent
sheaves on a weighted projective line, and the poset of elements in $W$ that
are below a Coxeter transformation in terms of a fixed partial order.
