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Register a new userA new proof of Gromovs theorem on groups of polynomial growth
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Authors
Bruce Kleiner
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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.
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We will present a new proof for the Gromovs theorem on almost flat manifolds ([Gr], [Ru]).
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
commensurable to some simply connected solvable Lie group S, the Lie shadow of G. We also show that large balls in G have an asymptotic shape, i.e. after a suitable renormalization, they converge to a limiting compact set which can be interpreted geometrically. We then discuss the speed of convergence, treat some examples and give an application to ergodic theory. We also answer a question of Burago about left invariant metrics and recover some results of Stoll on the irrationality of growth series of nilpotent groups.
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 exhibit a regular language of geodesics for a large set of elements of $BS(1,n)$ and show that the growth rate of this language is the growth rate of the group. This provides a straightforward calculation of the growth rate of $BS(1,n)$, which was initially computed by Collins, Edjvet and Gill in [5]. Our methods are based on those we develop in [8] to show that $BS(1,n)$ has a positive density of elements of positive, negative and zero conjugation curvature, as introduced by Bar-Natan, Duchin and Kropholler in [1].
This note is concerned with the geometric classification of connected Lie groups of dimension three or less, endowed with left-invariant Riemannian metrics. On the one hand, assembling results from the literature, we give a review of the complete classification of such groups up to quasi-isometries and we compare the quasi-isometric classification with the bi-Lipschitz classification. On the other hand, we study the problem whether two quasi-isometrically equivalent Lie groups may be made isometric if equipped with suitable left-invariant Riemannian metrics. We show that this is the case for three-dimensional simply connected groups, but it is not true in general for multiply connected groups. The counterexample also demonstrates that `may be made isometric is not a transitive relation.
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