The repetition threshold for binary rich words

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+root 2/2 (approximate to 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 + root 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 + root 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.