  Let Gamma be a fixed hyperbolic group. The Gamma-limit groups of Sela are
exactly the finitely generated, fully residually Gamma groups. We give a new
invariant of Gamma-limit groups called Gamma-discriminating complexity and show
that the Gamma-discriminating complexity of any Gamma-limit group is
asymptotically dominated by a polynomial. Our proof relies on an embedding
theorem of Kharlampovich-Myasnikov which states that a Gamma-limit group embeds
in an iterated extension of centralizers over Gamma. The result then follows
from our proof that if G is an iterated extension of centralizers over Gamma,
the G-discriminating complexity of a rank n extension of a cyclic centralizer
of G is asymptotically dominated by a polynomial of degree n.
