A finitary version of Gromov’s polynomial growth theorem.

*(English)*Zbl 1262.20044Let \(G\) be a finitely generated group with fixed finite symmetric generating set \(S\). The set \(B_S(R)\) is \(R\)-ball centered at the identity in the \(S\)-word metric. M. Gromov [Publ. Math., Inst. Hautes Étud. Sci. 53, 53-78 (1981; Zbl 0474.20018)] showed that if \(|B_S(R)|\leq R^d\) for some fixed \(d\) and all sufficiently large \(R\), then \(G\) is virtually nilpotent. This result was later generalized by L. van den Dries and A. J. Wilkie [J. Algebra 89, 349-374 (1984; Zbl 0552.20017)] who proved the result assuming only that the condition \(B_S(R)\leq R^d\) holds at infinitely many scales (instead of all scales). Later, B. Kleiner [J. Am. Math. Soc. 23, No. 3, 815-829 (2010; Zbl 1246.20038)] used different techniques to prove van den Dries’ and Wilkie’s result. The paper under review is a significant strengthening of this theorem that requires the growth condition to be satisfied at only one scale.

More specifically, the authors show that for some explicit constant \(C\) and for every finitely generated group \(G\) and \(d>0\) if there exists an \(R_0>\exp(\exp(Cd^C))\) for which the number of elements in a ball of radius \(R_0\) is bounded by \(R_0^d\), then \(G\) is virtually nilpotent. They also give a bound on the nilpotency degree. This remarkable result also yields a new result for finite groups. Additionally, they show that groups with slightly super-polynomial growth (where the bound is \(R^{c(\log\log R)^c}\)) are virtually nilpotent. Another interesting corollary can be expressed by saying that polynomial growth at one scale implies polynomial growth at all scales.

The proof of the main theorem follows the strategy used by Kleiner, but each step of the proof requires a great deal of work; it is not simply a rehashing of Kleiner’s proof. The authors give a comprehensive introduction that includes a plan of the proof, examples and comparisons with other work as well as all necessary definitions.

More specifically, the authors show that for some explicit constant \(C\) and for every finitely generated group \(G\) and \(d>0\) if there exists an \(R_0>\exp(\exp(Cd^C))\) for which the number of elements in a ball of radius \(R_0\) is bounded by \(R_0^d\), then \(G\) is virtually nilpotent. They also give a bound on the nilpotency degree. This remarkable result also yields a new result for finite groups. Additionally, they show that groups with slightly super-polynomial growth (where the bound is \(R^{c(\log\log R)^c}\)) are virtually nilpotent. Another interesting corollary can be expressed by saying that polynomial growth at one scale implies polynomial growth at all scales.

The proof of the main theorem follows the strategy used by Kleiner, but each step of the proof requires a great deal of work; it is not simply a rehashing of Kleiner’s proof. The authors give a comprehensive introduction that includes a plan of the proof, examples and comparisons with other work as well as all necessary definitions.

Reviewer: Gregory C. Bell (Greensboro)

##### MSC:

20F65 | Geometric group theory |

20F19 | Generalizations of solvable and nilpotent groups |

20F69 | Asymptotic properties of groups |

31C05 | Harmonic, subharmonic, superharmonic functions on other spaces |

##### Keywords:

groups of polynomial growth; virtually nilpotent groups; finitely generated groups; harmonic functions##### References:

[1] | Colding T., Minicozzi W. II: Harmonic functions on manifolds. Ann. Math. 146, 725–747 (1997) · Zbl 0928.53030 |

[2] | Dobrowolski E.: On a question of Lehmer and the number of irreducible factors of a polynomial. Acta Arith. 34(4), 391–401 (1979) · Zbl 0416.12001 |

[3] | van den Dries L., Wilkie A.J.: Gromov’s theorem on groups of polynomial growth and elementary logic. J. Alg. 89, 349–374 (1984) · Zbl 0552.20017 |

[4] | van den Dries L., Wilkie A.J.: An effective bound for groups of linear growth. Arch. Math. (Basel) 42(5), 391–396 (1984) · Zbl 0567.20016 |

[5] | Grigorchuk R.I.: On the Hilbert–Poincaré series of graded algebras that are associated with groups (Russian). Mat. Sb. 180(2), 207–225 (1989) |

[6] | R.I. Grigorchuk, Degrees of growth of finitely generated groups and the theory of invariant means, Math. USSR Izv. 25:2 (1985), 259–300; Russian original: Izv. Akad. Nauk SSSR Sr. Mat. 48:5 (1984), 939–985. |

[7] | Gromov M.: Groups of polynomial growth and expanding maps. Inst. Hautes Études Sci. Publ. Math. 53, 53–73 (1981) · Zbl 0474.20018 |

[8] | John F.: Extremum problems with inequalities as subsidiary conditions, Studies and Essays presented to R. Courant on his 60th birthday, pp. 187–204. Interscience Publishers Inc., New York, NY (1948) |

[9] | Kapovich M.: Hyperbolic Manifolds and Discrete Groups, Progress in Mathematics 183. Birkhäuser, Boston MA (2001) · Zbl 0958.57001 |

[10] | Kitaev A.Y., Shen A.H., Vyulyi M.N.: Classical and Quantum Computation, Graduate Studies in Mathematics 47. American Mathematical Society, Rhode Island (2002) |

[11] | B. Kleiner, A new proof of Gromov’s theorem on groups of polynomial growth, Jour. of the AMS, to appear. · Zbl 1246.20038 |

[12] | Korevaar N., Schoen R.: Global existence theorems for harmonic maps to nonlocally compact spaces. Comm. Anal. Geom. 5, 333–387 (1996) · Zbl 0908.58007 |

[13] | Kronecker L.: Zwei Sätze über Gleichungen mit ganzzahligen Coefficienten, J. für riene und angew. Math. 53, 173–175 (1857) · ERAM 053.1389cj |

[14] | Lazard M.: Groupes analytiques p-adiques. Inst. Hautes Études Sci. Publ. Math. 26, 389–603 (1965) · Zbl 0139.02302 |

[15] | J. Lee, Y. Makarychev, Eigenvalue multiplicity and volume growth, preprint. |

[16] | J. Lee, Y. Peres, Harmonic maps on amenable groups and a diffusive lower bound for random walks, preprint. · Zbl 1284.05250 |

[17] | Li P.: Harmonic sections of polynomial growth. Math. Res. Lett. 4, 35–44 (1997) · Zbl 0880.53039 |

[18] | Lubotzky A., Mann A.: Powerful p-groups. II. p-adic analytic groups. J. Algebra 105(2), 506–515 (1987) · Zbl 0626.20022 |

[19] | Lubotzky A., Mann A., Segal D.: Finitely generated groups of polynomial subgroup growth. Israel J. Math. 82(1), 363–371 (1993) · Zbl 0811.20027 |

[20] | Lubotzky A., Pyber L., Shalev A.: Discrete groups of slow subgroup growth. Israel J. Math. 96(B), 399–418 (1996) · Zbl 0877.20019 |

[21] | Mal’cev A.I.: Nilpotent torsion-free groups (Russian). Izv. Akad. Nauk SSSR. Ser. Mat. 13, 201–212 (1949) |

[22] | Mal’cev A.I.: Two remarks on nilpotent groups (Russian). Mat. Sb. N.S. 79, 567–572 (1955) |

[23] | Milnor J.: Growth of finitely generated solvable groups. J. Diff. Geom. 2, 447–449 (1968) · Zbl 0176.29803 |

[24] | Milnor J.: A note on curvature and fundamental group. J. Diff. Geom. 2, 1–7 (1968) · Zbl 0162.25401 |

[25] | N. Mok, Harmonic forms with values in locally compact Hilbert bundles, Proceedings of the Conference in Honor of Jean-Pierre Kahane (Orsay 1993), Special Issue (1995), 433–454. · Zbl 0891.58001 |

[26] | Segal D.: The finite images of finitely generated groups. Proc. London Math. Soc. (3) 82(3), 597–613 (2001) · Zbl 1022.20011 |

[27] | Shalom Y.: The growth of linear groups. J. Algebra 199(1), 169–174 (1998) · Zbl 0892.20023 |

[28] | Y. Shalom, T. Tao, On Kleiner’s proof of Gromov’s theorem and Lipschitz harmonic functions, in preparation. |

[29] | T. Tao, Product set estimates for non-commutative groups, preprint. · Zbl 1254.11017 |

[30] | T. Tao, Freiman’s theorem for solvable groups, preprint. · Zbl 1332.11015 |

[31] | Tits J.: Free subgroups in linear groups. J. Algebra 20, 250–270 (1972) · Zbl 0236.20032 |

[32] | Wolf J.: Growth of finitely generated solvable groups and curvature of Riemannian manifolds. J. Diff. Geom. 2, 421–446 (1968) · Zbl 0207.51803 |

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