# Poisson Furstenberg Boundary of topological groups, reference request

I'm trying to understand the relations between between the following group properties, in the case of (say, compactly generated locally compact) topological groups:

1. Group growth.
2. Amenability.
3. Poisson Furstenberg boundary.

I know the relations between these properties in the case of finitely generated groups (e.g. non-amenability implies both exponential growth and a non-trivial PF boundary)

Can anyone provide me with a reference?

Thanks!

• Subexponential growth implies amenable, by the same trivial argument as in the discrete case. – YCor Mar 9 '17 at 14:50
• there's no Poisson-Furstenberg boundary for $G$, but for a pair $(G,\mu)$, $\mu$ being a probability measure. – YCor Mar 10 '17 at 13:48

Edit: I found the introduction of this article of A. Furman (mainly quoting results of Furstenberg) useful for one approach to non-amenability, although it maybe isn't quite what you are looking for:

http://homepages.math.uic.edu/~furman/preprints/fb.pdf

In particular, you can define the universal boundary (boundary = compact Hausdorff $G$-space such that the action is minimal and strongly proximal) for any locally compact group $G$, and $G$ is amenable if and only if the universal boundary is trivial. It looks like this universal boundary $B(G)$ is what is meant by the Furstenberg boundary of a locally compact group.

Given a random walk on $G$ that generates the group, one can then define its Poisson(-Furstenberg) boundary. Depending on the measure, this can be larger than $B(G)$. If I am reading Furstenberg's paper correctly, for a semisimple Lie group they can all be realized as covers of $B(G)$, so $B(G)$ is the smallest Poisson boundary; not sure how far this generalizes.

• I'm sceptical about "It looks like this universal boundary is what is usually meant by the Poisson-Furstenberg boundary of a locally compact group"; as far as I know these are distinct objects. Look e.g. at Erschler's survey on Poisson-Furstenberg boundaries (mathunion.org/ICM/ICM2010.2/Main/icm2010.2.0681.0704.pdf) or Hartman's (math.northwestern.edu/~hartman/pdfs/FPB-course.pdf) – YCor Mar 10 '17 at 13:49
• @YCor is definitely right. People tend to agree nowadays on the following terminology: 1) given a locally compact group $G$, its maximal strongly proximal minimal compact action is termed "The Furstenberg Boundary of $G$" and – Uri Bader Mar 10 '17 at 15:58
• 2) given a locally compact second countable $G$ and a probability measure $\mu$ on $G$, the unique $G$-measured space $(B,\nu)$ where $\nu$ satisfies $\mu*\nu=\nu$ and for which the "Poisson Transform" $L^\infty(B)\to H^{\infty}(G)$ taking $f$ to $g\mapsto \int fdg\nu$ is an isometry onto the space of $\mu$-harmonic functions on $G$, is called "The Furstenberg-Poisson Boundary of $(G,\mu)$". – Uri Bader Mar 10 '17 at 16:01
• Both objects were defined by Furstenberg. The second one was originally given (by Furstenberg) the name "Poisson Boundary", and this is also how it is called in Furman's survey, but as I mentioned, it is now called mainly the "Furstenberg-Possion Boundary". Admittedly, confusing terminology, but too late to be changed. – Uri Bader Mar 10 '17 at 16:03
• Colin, I shall add: contributing to the confusion is the fact that for a semisimple Lie group $G$ with a minimal parabolic $P$: 1) the Furstenberg Boundary is $G$-homeomorphic to $G/P$ and 2) for any reasonable $\mu$ the Furstenberg-Poisson boundary could be realized as well on $G/P$, that is it is isomorphic to $(G/P,\nu)$ as a $G$-Lebesgue space for some measure $\nu$ (which depends on $\mu$). – Uri Bader Mar 10 '17 at 16:52

In their paper "Existence of positive harmonic functions on groups and on covering manifolds", Bougerol & Elie give an overview of the connection between the three properties growth, amenability, and existence of bounded/positive harmonic functions.

In short, under the right conditions, amenability is equivalent to the existence of (non-trivial) bounded harmonic functions, polynomial growth implies no positive harmonic functions, and exponential growth implies existence of positive (non-trivial) harmonic function.

More accurately:

1. [Azencott] A non-amenable compactly generated locally compact group equipped with an adapted probability measure admit non-constant bounded (hence also positive) harmonic function.

2. [Guivarc'h, Kaimanovich, Alexopoulos] A connected amenable compactly generated locally compact group equipped with an adapted, centered, with a compactly supported continuous density probability measure, admits no bounded harmonic functions other than the constants.

3. [Hebish & Saloff Coste] A compactly generated locally compact group with polynomial growth equipped with an adapted, symmetric, with a compactly supported continuous density probability measure, admits no positive harmonic functions other than the (positive) constants.

Bougerol & Elie then prove:

1. A compactly generated locally compact group that admits a continuous homomorphism into a almost connected group and such that the closure of the image has exponential growth, equipped with an adapted, centered, with a continuous density probability measure that has a third moment, admits a non-constant positive harmonic function.

Most of the references are in French, so no wonder I didn't find them before. :)

I would appreciate any comments on this, and I thank all the respondents!