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Abstract: The most important invariant of a finite dimensional algebra is its dimension. Let \(A\) be a finitely generated, infinite dimensional associative or Lie algebra over some base field \(F\). A useful way to 'measure its infinitude' is to study its growth rate, namely, the asymptotic behavior of the dimensions of the spaces spanned by (at most \(n\))-fold products of some fixed generators. Up to a natural asymptotic equivalence relation, this function becomes a well-defined invariant of the algebra itself, independent of the specification of generators.

The question of 'how do algebras grow?', or, which functions can be realized as growth rates of algebras (perhaps with additional algebraic properties, as grading, simplicity etc.) plays an important role in classifying infinite dimensional algebras of certain classes, and is thus connected to ring theory, noncommutative projective geometry, quantum algebra, arithmetic geometry, combinatorics of infinite words, symbolic dynamics and more.

We present new results on possible and impossible growth rates of important classes of associative and Lie algebras, thereby settling several open questions in this area. Among the tools we apply are novel techniques and recent constructions arising from noncommutative algebra, combinatorics of (infinite trees of) infinite words and convolution algebras of étale groupoids attached to them.