Background: A generating set of a group G is a subset such that every can be written as a finite product of elements of S and their inverses. Group G is called finitely generated if it has a finite generating set S. Let be the number of elements of G which are the product of at most n elements in . We say that group G (i) has a polynomial growth rate if for some constants ; (ii) has an exponential growth rate if for some , and (iii) is of intermediate growth if neither (i) nor (ii) is true. Properties (i)-(iii) does not depend on S.
The Theorem: On 6th January 2016, Volodymyr Nekrashevych submitted to arxiv a paper in which he, among other results, proved the existence of finitely generated simple groups of intermediate growth.
Short context: The rate of growth of can tell a lot about structure of the underlying group G, and is one of the central research directions in group theory. The existence of groups of intermediate growth was an open question, until the first examples were constructed by Grigorchuk in 1980. Since then, other constructions have been discovered, but all known groups of intermediate growth were not simple. The Theorem provides the first examples of simple groups of intermediate growth, affirnatively answering the 1984 question of Grigorchuk. In particular, the Theorem generalises (and implies) this previous result about existence of finitely generated simple amenable groups (because every group of intermediate growth is amenable).
Links: Free arxiv version of the original paper is here, journal version is here.
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