A Finitary Version of Gromov’s Polynomial Growth Theorem

By Yehuda Shalom, Terence Tao. 2009

108 citations recorded by the source corpus.

We show that for some absolute (explicit) constant C, the following holds for every finitely generated group G, and all d > 0: If there is someR0 > exp(exp(CdC)) for which the number of elements in a ball of radius R0 in a Cayley graph of G is bounded by $${R_0^d}$$ , then G has a finite-index subgroup which is nilpotent (of step < Cd). An effective bound on the finite index is provided if “nilpotent” is replaced by “polycyclic”, thus yielding a non-trivial result for finite groups as well.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/0a7517f0de3ee61e6ed1a36c6f21ba2849a8eeab

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In Discrete Mathematics and Combinatorics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

The structure of approximate groups (2011, background): “On the other hand, the results in [37, 50], which follow a strategy close to that of Kleiner’s work [36], yield more effective quantitative control on the index and step of G′, especially in the case when S is of bounded cardinality.”
Milnor's Problem on the Growth of Groups and its Consequences (2011, background): “In [203], a pair (G,A) (a group G with a finite generating set A) is called an (R0, d)-growth group if”
Local Universality of Zeroes of Random Polynomials (2013, methodology): “Indeed, if we apply [Shalom & Tao, 2010, Proposition 13.1], we conclude that there exists a non-zero vector ~c ∈ Zd and a natural number k = O(1) such that T k1 ~c = ~c.”