In his paper "On discrete subgroups of the two by two projective linear group over $\mathfrak{p}$-adic fields", Yasutaka Ihara considers an abstract group $G$ together with a length function $l$ from $G$ into the non-negative integers with the following properties.


For each non-negative integer $l$, denote by $G\_{l}$ the set $\{x\in G\mid l(x)=l\}$. $U=G\_{0}$ is a subgroup of $G$ and $G\_{l}^{-1}=G\_{l}$, $UG\_{l}U=G\_{l}$, $|U\setminus G\_{l}|<\infty$. 


He presents one more axiom later, but after presenting this first axiom he says that this is sufficient for us to be able to consider the ring of double sets of $U$ in $G$. Presumably in order for this to be a ring the product of two double cosets must be a union of finitely many double cosets. I was wondering how this follows from the information given so far.



Existence of the double coset ring on paper of Ihara
Given $x\in G$, then $x\in G\_l$ for some $l$.
Then $U\backslash UxU\subset U\backslash UG\_lU=U\backslash G\_l$. The latter is a finite set.


