Let $G$ be group which acts on a space $X$. The group is said to act transitively on $X$ if it admits exactly one orbit. The subset of points in $G$ which leave a point $x \in X$ fixed form the so-called isotropy group of stabiliser. I am confused by the meaning of the following sentence:
The group $G$ acts on $X$ transitively with compact isotropy.
Is this to be read as the isotropy group is compact, or the quotient of $X$ by the isotropy group is compact? The question is motivated by a reading of Griffiths-Schmid's paper on locally homogeneous spaces.