  The purpose of this paper is to study the properties of the irrational-slope
Thompson's group $F_\tau$ introduced by Cleary in 1995. We construct
presentations, both finite and infinite and we describe its combinatorial
structure using binary trees. We show that its commutator group is simple.
Finally, inspired by the case of Thompson's group F, we define a unique normal
form for the elements of the group and study the metric properties for the
elements based on this normal form. As a corollary, we see that several
embeddings of $F$ in $F_\tau$ are undistorted.
