We study the avoidance of Abelian powers of words and consider three reasonablegeneralizations of the notion of Abelian power to fractional powers. Our main goal is tofind an Abelian analogue of the repetition threshold, i.e., a numericalvalue separating k-avoidable and k-unavoidable Abelianpowers for each size k of the alphabet. We prove lower bounds for theAbelian repetition threshold for large alphabets and all definitions of Abelian fractionalpower. We develop a method estimating the exponential growth rate of Abelian-power-freelanguages. Using this method, we get non-trivial lower bounds for Abelian repetitionthreshold for small alphabets. We suggest that some of the obtained bounds are the exactvalues of Abelian repetition threshold. In addition, we provide upper bounds for thegrowth rates of some particular Abelian-power-free languages.

The repetition threshold introduced by Dejean and Brandenburg is the smallest real number α such that there exists an infinite word over a k-letter alphabet that avoids β-powers for all β > α. We extend this notion to colored graphs and obtain the value of the repetition thresholds of trees and “large enough” subdivisions of graphs for every alphabet size.