Let R be an extended affine root system in a vector space V. The Weyl
group of R is the group generated by reflections based on nonisotropic
elements of R. Recently, we found a presentation for the Weyl group of any
extended affine root system of type $A_1$, which is very similar to the
Coxeter presentation of finite and affine Weyl groups. In this talk, we first
discuss a geometric proof of the mentioned presentation and then we talk about
a length function on these groups. In the geometric proof, we use paths and
loops created by simplices. Loops defined in this way are in a one-to-one
correspondence with the relations of the Weyl group. On the other hand, one
can define a way to reduce each loop to a "fundamental" loop. These "fundamental"
loops are in a one-to-one correspondence with the relations of the Weyl group
which we consider for our presentation.
We know that Weyl groups of affine Kac-Moody root systems are both Coxeter groups
and Weyl groups of extended affine root systems of nullity one. Furthermore, each
Coxeter groupi $W$ has a length function which for any element $w \in W$ gives the
minimum number of Coxeter generators needed to build $w$. On the other hand, we
showed that the Weyl groups of extended affine root systems of type $A_1$ have a
"similar" presentation to the affine Coxeter presentation. In order to show that
Coxeter groups and these Weyl groups are "related", in the second part of this talk,
we discuss a length fucntion for Weyl groups of extended affine root systems of type
$A_1$ which is the same as the Coxeter group's length function for Weyl groups of affine
Kac-Moody root systems of type $A_1$. Our proof of the existence of this length function
uses algebraic, combinatorial and geometric properties of these class of Weyl groups and
their underlying root systems. For instance, we define a notion of positive and negative
roots on extended affine root systems which is highly connected to the length function
of the corresponding Weyl groups.