Date:
Thu, 26/03/202012:00-13:00
Location:
Seminar room 209, Manchester Building, Jerusalem, Israel.
How do algebras grow?
The question of `how do algebras grow?', or, which functions can be realized as growth functions of algebras (associative/Lie, or algebras having certain additional algebraic properties) is a major problem in the meeting point of several mathematical fields including algebra, combinatorics, symbolic dynamics and more.
For finitely generated groups the situation is rather rigid: Gromov proved in 1981 that finitely generated groups of polynomial growth are virtually nilpotent. The Grigorchuk group and far reaching generalizations or variations constructed upon it provide examples of groups with intermediate (namely, super-polynomial but subexponential) growth, but still much is unknown about what precisely is the possible behavior of growth functions of groups.
For finitely generated associative algebras, a much wider class of growth functions is possible. Any increasing and submultiplicative function is - up to an important polynomial factor - the growth function of a finitely generated algebra; nil algebras (which are analogs of Burnside groups) of polynomial growth exist; any `sufficiently regular' growth function which is more rapid than $n^{\log n}$ is the growth function of a simple algebra (simple groups with intermediate growth were only recently constructed); etc.
We present new results on possible (and impossible) growth functions of important classes of algebras (and Lie algebras), by combining novel techniques and constructions from noncommutative algebra, combinatorics of infinite words and theory of (convolution algebras of) étale groupoids attached to them. We derive answers to several open questions posed by experts in the area and survey the main open problems in the field.
The question of `how do algebras grow?', or, which functions can be realized as growth functions of algebras (associative/Lie, or algebras having certain additional algebraic properties) is a major problem in the meeting point of several mathematical fields including algebra, combinatorics, symbolic dynamics and more.
For finitely generated groups the situation is rather rigid: Gromov proved in 1981 that finitely generated groups of polynomial growth are virtually nilpotent. The Grigorchuk group and far reaching generalizations or variations constructed upon it provide examples of groups with intermediate (namely, super-polynomial but subexponential) growth, but still much is unknown about what precisely is the possible behavior of growth functions of groups.
For finitely generated associative algebras, a much wider class of growth functions is possible. Any increasing and submultiplicative function is - up to an important polynomial factor - the growth function of a finitely generated algebra; nil algebras (which are analogs of Burnside groups) of polynomial growth exist; any `sufficiently regular' growth function which is more rapid than $n^{\log n}$ is the growth function of a simple algebra (simple groups with intermediate growth were only recently constructed); etc.
We present new results on possible (and impossible) growth functions of important classes of algebras (and Lie algebras), by combining novel techniques and constructions from noncommutative algebra, combinatorics of infinite words and theory of (convolution algebras of) étale groupoids attached to them. We derive answers to several open questions posed by experts in the area and survey the main open problems in the field.