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We generalize both the notion of polynomial functions on Lie groups and the notion of horizontally affine maps on Carnot groups. We fix a subset $S$ of the algebra $mathfrak g$ of left-invariant vector fields on a Lie group $mathbb G$ and we assume that $S$ Lie generates $mathfrak g$. We say that a function $f:mathbb Gto mathbb R$ (or more generally a distribution on $mathbb G$) is $S$-polynomial if for all $Xin S$ there exists $kin mathbb N$ such that the iterated derivative $X^k f$ is zero in the sense of distributions. First, we show that all $S$-polynomial functions (as well as distributions) are represented by analytic functions and, if the exponent $k$ in the previous definition is independent on $Xin S$, they form a finite-dimensional vector space. Second, if $mathbb G$ is connected and nilpotent we show that $S$-polynomial functions are polynomial functions in the sense of Leibman. The same result may not be true for non-nilpotent groups. Finally, we show that in connected nilpotent Lie groups, being polynomial in the sense of Leibman, being a polynomial in exponential chart, and the vanishing of mixed derivatives of some fixed degree along directions of $mathfrak g$ are equivalent notions.
We give a new proof of Gromovs theorem that any finitely generated group of polynomial growth has a finite index nilpotent subgroup. Unlike the original proof, it does not rely on the Montgomery-Zippin-Yamabe structure theory of locally compact groups.
We get asymptotics for the volume of large balls in an arbitrary locally compact group G with polynomial growth. This is done via a study of the geometry of G and a generalization of P. Pansus thesis. In particular, we show that any such G is weakly
We consider harmonic functions of polynomial growth of some order $d$ on Cayley graphs of groups of polynomial volume growth of order $D$ w.r.t. the word metric and prove the optimal estimate for the dimension of the space of such harmonic functions.
We prove that any Cayley graph $G$ with degree $d$ polynomial growth does not satisfy ${f(n)}$-containment for any $f=o(n^{d-2})$. This settles the asymptotic behaviour of the firefighter problem on such graphs as it was known that $Cn^{d-2}$ firefig
We show that, in compact semisimple Lie groups and Lie algebras, any neighbourhood of the identity gets mapped, under the commutator map, to a neighbourhood of the identity.