On cogrowth function of algebras and its logarithmical gap

Abstract

Let $A\cong k⟨X⟩/I$ be an associative algebra. A finite word over alphabet $X$ is $I${\it-reducible} if its image in $A$ is a $k$-linear combination of length-lexicographically lesser words. An {\it obstruction} in a subword-minimal $I$-reducible word. A {\em cogrowth} function is number of obstructions of length $\le n$. We show that the cogrowth function of a finitely presented algebra is either bounded or at least logarithmical. We also show that an uniformly recurrent word has at least logarithmical cogrowth.

arXiv comment5 pages

References in corpus (1)

• The maximal length of the period of a periodic word defined by restrictions