Let $K$ be a global field with adele group $A_K$, then $GL_2(A_K)/Z(A_K)GL_2(K)$ is not compact, where $Z(A_K)$ is the center of $GL_2(A_K)$, the multiples of identity.
When considering $A_K/K$, we consider a set of representatives which lies in a compact space; when considering $GL_2(A_K)/GL_2(K)$, we consider $\|\cdot\|\circ\det$ whose image is not compact. But how about $GL_2(A_K)/Z(A_K)GL_2(K)$? I can neither find a noncompact fundamental domain nor construct a continuous function on it.