# Who proved that a group of polynomial growth has growth exactly polynomial?

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 http://de.arxiv.org/PS_cache/arxiv/pdf/1011/1011.5266v2.pdf, 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?

Valerio

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A good start is Wikipedia: en.wikipedia.org/wiki/… –  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 front.math.ucdavis.edu/1012.1325, 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