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Gaps and approximations in the space of growth functions

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Abstract

An important problem in combinatorial noncommutative algebra is to give an asymptotic characterization the growth functions of finitely generated algebras (equivalently, semigroups, or hereditary languages). The growth function of every finitely generated, infinite-dimensional algebra is increasing and submultiplicative. The question of to what extent these natural necessary conditions are also sufficient—and in particular, whether they are sufficient at least for sufficiently rapid functions—was posed and studied by various authors and has attracted a flurry of research. While every increasing and submultiplicative function is realizable as a growth function up to a linear error term, we show that there exist arbitrarily rapid increasing submultiplicative functions which are not equivalent to the growth of any algebra, thus resolving the aforementioned problem and settling a question posed by Zelmanov (and repeated by Alahmadi–Alsulami–Jain–Zelmanov). These can be interpreted as ‘holes’ in the space of growth functions, accumulating to exponential functions in the order topology. We show that there exist monomial algebras and hereditary languages whose growth functions encode the existence of non-prolongable words, and algebras whose growth functions encode the existence of nilpotent ideals (in the graded case). This negatively solves another conjecture of Alahmadi–Alsulami–Jain–Zelmanov in the graded case.

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Notes

  1. Here \(\alpha =\log 2/\log \eta \approx 0.7674\), where \(\eta \) is the positive root of \(X^3-X^2-2X-4\). ‘Uniformly faster’ here means that \(f(2n)\le f(n)^2\le f(\eta n)\) for \(n\gg 1\).

  2. Even though it is a priori possible that any sufficiently rapid, increasing and submultiplicative function is equivalent to a function which additionally satisfies \(f'(n)\ge n+1\).

  3. The growth of any algebra is equivalent to the growth of some graded algebra; it is potentially possible, however, that a function is equivalent to the growth of a primitive algebra, but not equivalent to the growth of any primitive graded algebra.

  4. In the more general setting of dynamical systems, transitivity may refer to having any non-empty open subset intersect any other non-empty open subset, after a suitable shift.

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Acknowledgements

The author thanks Jason Bell and Efim Zelmanov for related inspiring discussions and Guy Blachar for helpful suggestions regarding the paper. The author is thankful to the referee for their careful reading of the paper and useful comments.

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  1. Department of Mathematics, University of Washington, Seattle, WA, 98195-4350, USA

    Be’eri Greenfeld

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Greenfeld, B. Gaps and approximations in the space of growth functions. Sel. Math. New Ser. 29, 62 (2023). https://doi.org/10.1007/s00029-023-00862-x

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