A new proof of Gromov's theorem on groups of polynomial growth

By Bruce Kleiner. 2007

151 citations recorded by the source corpus.

We give a new proof of Gromov’s theorem that any finitely generated group of polynomial growth has a finite index nilpotent subgroup. The proof does not rely on the MontgomeryZippin-Yamabe structure theory of locally compact groups.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/309cc6afb0f65325c7f990a65c04ad036ddcd7e7

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Citations in the corpus

Combinatorial algebra : syntax and semantics (2014, unknown): “Fortunately, there are detailed proofs of his results in [322, 219, 91, 181] (the proof in [128] is not long and is not that hard to read also).”
The structure of approximate groups (2011, result): “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.”
A Finitary Version of Gromov’s Polynomial Growth Theorem (2009, methodology): “Statement (iv), established in section 9, was proven in [ Kl ] (as in Gromov’s original [Gr]) using the Tits alternative, or by a weaker variant of that alternative due to the first author [Sh].”
Equiangular lines with a fixed angle (2019, methodology): “Following the approach of Colding and Minicozzi [5] on harmonic functions on manifolds and Kleiner’s proof [15] of Gromov’s theorem on groups of polynomial growth [11], Lee and Makarychev [16] showed that in groups with bounded doubling constant K = maxR>0 |B(2R)| /|B(R)| (where B(R) is the ball of ”
Milnor's Problem on the Growth of Groups and its Consequences (2011, background): “(Kleiner 2010) Let G be a group of weakly polynomial growth, and l ∈ (0,∞).”
Probability on Trees and Networks (2017, background): “In case (i), the assumption that supn Vn/n(4) = ∞ implies that Vn ≥ cn5 for some c > 0 and all n: see Kleiner (2010). By Corollary 6.32, this yields Px[Xn = x] ≤ Cn−5/2. Thus the sum in (10.32) converges. In case (ii), combining the results (14.5), (14.12), and (14.19) in Woess (2000), we infer that”
Nilpotent Group C*-algebras as Compact Quantum Metric Spaces (2002, background): “there exists C<∞ such that L≤ C−1L˜, that is, the balls B˜(r) associated to L˜ satisfy B˜(r) ⊂ B(Cr). Thus if Lhas polynomial growth, it follows that L˜ does also. According to a theorem of Gromov [8, 10, 11, 25], this implies that Gis nilpotent-by-ﬁnite. But the property of strong polynomial gro”