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I need to put a reference about the classical result that a f.g. group of polynomial growth has growth which is exactly polynomial.

Talking personally with people and also here in A question about groups of intermediate growth, it seems that the result is attributed to Gromov and Pansu. In particular Gromov proved that a group of polynomial growth is virtually nilpotent and Pansu proved, in a paper in Ergodic Th & Dyn Systems that goes back to 1983, that a nilpotent group has exactly polynomial growth (is this right?)

Now, reading the introduction of, Bartholdi and Erschler say that the result is due to Guivarch (without reference) and Bass (in 1972).

So I am now little confused... who proved this theorem?

I guess that the problem is that nobody proved that theorem in this form. In this case, can anybody tell me a short history of the result, in such a way not to write wrong things?

Thank you in advance,


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A good start is Wikipedia:… – Alain Valette Nov 22 '11 at 11:25
Igor's answer says who proved nilpotent groups have exactly polynomial growth. You then need Gromov to get polynomial growth gives virtual nilpotence. – Benjamin Steinberg Nov 22 '11 at 11:26
There are two questions. You can ask what is the best exponent of polynomial growth (it is non-trivial that it should be an integer!), and this points to Bass (for virtually nilpotent) and Guivarc'h (for Lie groups). And you can ask for a lower bound, i.e. is there a true polynomial equivalent for the growth function, and this points to Pansu. – Alain Valette Nov 22 '11 at 11:36
See, Section 3.2. – Mark Sapir Nov 22 '11 at 12:51
(after checking in Mark's paper): for nilpotent groups, Guivarch & Bass proved that the growth is equivalent to $n^d$, with an explicit $d$; and Pansu proved that the limit $\lim \frak{b(n)}{n^d}$ exists, implying that the sequence of ALL balls is F\olner. – Alain Valette Nov 22 '11 at 13:25

Guivarc'h certainly did prove such a result, see MR0302819 (46 #1962) Guivarc'h, Yves Groupes de Lie à croissance polynomiale. (French) C. R. Acad. Sci. Paris Sér. A-B 272 (1971), A1695–A1696. 22E15

As did Bass:

MR0379672 (52 #577) Bass, H. The degree of polynomial growth of finitely generated nilpotent groups. Proc. London Math. Soc. (3) 25 (1972), 603–614.

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