  We propose two geometric versions of the bounded reduction property and find
conditions for them to coincide. In particular, for the natural automatic
structure on a hyperbolic group, the two notions are equivalent. We study
endomorphisms with $L$-quasiconvex image and prove that those with finite
kernel satisfy a synchronous version of the bounded reduction property.
Finally, we use these techniques to prove $L$-quasiconvexity of the equalizer
of two endomorphisms under certain (strict) conditions.
