Geometry of Locally Compact Groups of Polynomial Growth and Shape of Large Balls

By Emmanuel Breuillard. 2007

113 citations recorded by the source corpus.

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. Pansu's 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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/cff2cb132c3be57a65690d658c35b1588cab8d8b

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Geometry and Topology

1. Simeng Wang
2. Yves Cornulier
3. Benben Liao
4. G. Hong

In Mathematical Physics

1. Simeng Wang
2. Yves Cornulier
3. Benben Liao
4. G. Hong
5. Assaf Naor
6. Yuval Peres

Citations in the corpus

• Noncommutative maximal ergodic inequalities associated with doubling conditions (2017, background): “ hand, in the classical ergodic theory, a number of significant developments related to individual ergodic theorems for group actions have been established in the past years. In particular, Breuillard [Bre14] and Tessera [Tes07] studied the balls in groups of polynomial growth; they proved that for a”
• Dimension of asymptotic cones of Lie groups (2007, methodology): “ymptotic cone is homeomorphic to Rd: in the essential case where P is nilpotent, this is due to Pansu [Pan], and in general this follows from the nilshadow construction (see for instance Lemma 2.2 or [Bre]). Finally, we have to know that Rd has covering dimension d[Nag, Theorem IV.4]. The second ste”