  Kida and Tucker-Drob recently extended the notion of inner amenability from
countable groups to discrete p.m.p. groupoids. In this article, we show that
inner amenable groupoids have "fixed priced 1" in the sense that every
principal extension of an inner amenable groupoid has cost 1. This
simultaneously generalizes and unifies two well known results on cost from the
literature, namely, (1) a theorem of Kechris stating that every ergodic p.m.p.
equivalence relation admitting a nontrivial asymptotically central sequence in
its full group has cost 1, and (2) a theorem of Tucker-Drob stating that inner
amenable groups have fixed price 1.
