On cogrowth function of algebras and its logarithmical gap

TL;DR

This paper investigates the growth behavior of obstructions in finitely presented associative algebras, demonstrating that the cogrowth function is either bounded or grows at least logarithmically, with implications for uniformly recurrent words.

Contribution

It establishes a dichotomy for the cogrowth function of finitely presented algebras and links it to properties of uniformly recurrent words, providing new insights into algebraic and combinatorial structures.

Findings

Cogrowth function is either bounded or at least logarithmic.

Uniformly recurrent words have at least logarithmic cogrowth.

Provides a new perspective on the growth behavior of obstructions in algebras.

Abstract

Let $A \cong k\langle X \rangle / I$ 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.