The growth of an infinite-dimensional algebra is a fundamental tool to measure its infinitude. Growth of noncommutative algebras plays an important role in noncommutative geometry, representation theory, differential algebraic geometry, symbolic dynamics and various recent homological stability results in number theory and arithmetic geometry.

We analyze the space of growth functions of algebras, answering a question of Zelmanov (2017) on the existence of certain 'holes' in this space, and prove a strong quantitative version of the Kurosh Problem on algebraic algebras. We use minimal subshifts with highly correlated oscillating complexities to resolve a question posed by Krempa-Okninski (1987) and Krause-Lenagan (2000) on the GK-dimension of tensor products.

An important property implied by subexponential growth (for both groups and algebras) is amenability. We show that minimal subshifts of positive entropy give rise to amenable graded algebras of exponential growth, answering a conjecture of Bartholdi (2007; naturally extending a wide open conjecture of Vershik on amenable group rings).

This talk is partially based on joint works with J. Bell and with E. Zelmanov.