  A word of length $n$ is rich if it contains $n$ nonempty palindromic factors.
An infinite word is rich if all of its finite factors are rich. Baranwal and
Shallit produced an infinite binary rich word with critical exponent
$2+\sqrt{2}/2$ ($\approx 2.707$) and conjectured that this was the least
possible critical exponent for infinite binary rich words (i.e., that the
repetition threshold for binary rich words is $2+\sqrt{2}/2$). In this article,
we give a structure theorem for infinite binary rich words that avoid
$14/5$-powers (i.e., repetitions with exponent at least 2.8). As a consequence,
we deduce that the repetition threshold for binary rich words is
$2+\sqrt{2}/2$, as conjectured by Baranwal and Shallit. This resolves an open
problem of Vesti for the binary alphabet; the problem remains open for larger
alphabets.
