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, homological stability results and more.

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-dimensions of tensor products.

An important property implied by subexponential growth (both for groups and for algebras) is amenability. We show that minimal subshifts of positive entropy give rise to graded algebras of exponential growth all of whose representations are amenable, thereby answering a conjecture of Bartholdi (2007; extending an open question of Vershik). Finally, we discuss amenability-type properties of associative and Lie algebras related to this question.

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