On cogrowth function of algebras and its logarithmical gap
Alexei Kanel-Belov, Igor Melnikov, Ivan Mitrofanov · Comptes Rendus Mathématique · 2021
Let A ≅ k 〈 X 〉 / I be an associative algebra. A finite word over alphabet X is I -reducible if its image in A is a k -linear combination of length-lexicographically lesser words. An obstruction is a subword-minimal I -reducible word. If the number of obstructions is finite then I has a finite Gröbner basis, and the word problem for the algebra is decidable. A cogrowth function is the number of obstructions of length ≤ 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.