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An amenable group $G$ acting by homeomorphisms on a compact topological space $X$ preserves a probability measure on $X$; in fact, one can given a definition of amenability in such terms. For example, if $G$ is finite, it preserves an atomic measure supported on any orbit. If $G = \mathbb{Z}$, one can take a sequence of almost invariant probability measures, supported on the subset $[-n,n] \cdot p$ (where $p \in X$ is arbitrary), and any weak limit will be invariant. For a general amenable group, in place of the subsets $[-n,n] \subset \mathbb{Z}$, one works with a sequence of Folner sets; i.e. subsets with the property that the ratio of their size to the size of their boundary goes to zero (so to speak).

But if $G$ is not amenable, it is generally not true that there is any probability measure on $X$ invariant under the action of $G$. The best one can expect is a probability measure which is invariant on average. Such a measure is called a harmonic measure (or a stationary measure) for the $G$-action on $X$. To be concrete, suppose $G$ is finitely generated by a symmetric generating set $S$ (symmetric here means that if $s \in S$, then $s^{-1} \in S$). Let $M(X)$ denote the space of probability measures on $X$. One can form an operator $\Delta:M(X) \to M(X)$ defined by the formula

$\Delta(\mu) = \frac {1} {|S|} \sum_{s \in S} s_*\mu$

and then look for a probability measure $\nu$ stationary under $\Delta$, which exists for quite general reasons. This measure $\nu$ is the harmonic measure: the expectation of the $\nu$-measure of $s(A)$ under a randomly chosen $s \in S$ is equal to the $\nu$-measure of $A$. Note for any probability measure $\mu$ that $s_*\mu$ is absolutely continuous with respect to $\Delta(\mu)$; in fact, the Radon-Nikodym derivative satisfies $ds_*\mu/d\Delta(\mu) \le |S|$. Substituting $\nu$ for $\mu$ in this formula, one sees that the measure class of $\nu$ is preserved by $G$, and that for every $g \in G$, we have $dg_*\nu/d\nu \le |S|^{|g|}$, where $|g|$ denotes word length with respect to the given generating set.

The existence of harmonic measure is especially useful when $X$ is one-dimensional, e.g. in the case that $X=S^1$. In one dimension, a measure (at least one of full support without atoms) can be “integrated” to a path metric. Consequently, any finitely generated group of homeomorphisms of the circle is conjugate to a group of bilipschitz homeomorphisms (if the harmonic measure associated to the original action does not have full support, or has atoms, one can “throw in” another random generator to the group; the resulting action can be assumed to have a harmonic measure of full support without atoms, which can be integrated to give a structure with respect to which the group action is bilipschitz). In fact, Deroin-Kleptsyn-Navas showed that any countable group of homeomorphisms of the circle (or interval) is conjugate to a group of bilipschitz homeomorphisms (the hypothesis that $G$ be countable is essential; for example, the group $\mathbb{Z}^{\mathbb{Z}}$ acts in a non-bilipschitz way on the interval — see here).

Suppose now that $G = \pi_1(M)$ for some manifold $M$. The action of $G$ on $S^1$ determines a foliated circle bundle $S^1 \to E \to M$; i.e. a circle bundle, together with a codimension one foliation transverse to the circle fibers. To see this, first form the product $\widetilde{M} \times S^1$ with its product foliation by leaves $\widetilde{M} \times \text{point}$, where $\widetilde{M}$ denotes the universal cover of $M$. The group $G = \pi_1(M)$ acts on $\widetilde{M}$ as the deck group of the covering, and on $S^1$ by the given action; the quotient of this diagonal action on the product is the desired circle bundle $E$. The foliation makes $E$ into a “flat” circle bundle with structure group $\text{Homeo}^+(S^1)$. The foliation allows us to associate to each path $\gamma$ in $M$ a homeomorphism from the fiber over $\gamma(0)$ to the fiber over $\gamma(1)$; integrability (or flatness) implies that this homeomorphism only depends on the relative homotopy class of $\gamma$ in $M$. This identification of fibers is called the holonomy of the foliation along the path $\gamma$. If $M$ is a Riemannian manifold, there is another kind of harmonic measure on the circle bundle; in other words, a probability measure on each circle with the property that the holonomy associated to an infinitesimal random walk on $M$ preserves the expected value of the measure. This is (very closely related to) a special case of a construction due to Lucy Garnett which associates a harmonic transverse measure to any foliation $\mathcal{F}$ of a manifold $N$, by finding a fixed point of the leafwise heat flow on the space of probability measures on $N$, and disintegrating this measure into the product of the leafwise area measure, and a “harmonic” transverse measure.

In any case, we normalize our foliated circle bundle so that each circle has length $2\pi$ in its harmonic measure. Let $X$ be the vector field on the circle bundle that rotates each circle at unit speed, and let $\alpha$ be the $1$-form on $E$ whose kernel is tangent to the leaves of the foliation. We scale $\alpha$ so that $\alpha(X)=1$ everywhere. The integrability condition for a foliation is expressed in terms of the $1$-form as the identity $\alpha \wedge d\alpha = 0$, and we can write $d\alpha = -\beta \wedge \alpha$ where $\beta(X)=0$. More intrinsically, $\beta$ descends to a $1$-form on the leaves of the foliation which measures the logarithm of the rate at which the transverse measure expands under holonomy in a given direction (the leafwise form $\beta$ is sometimes called the Godbillon class, since it is “half” of the Godbillon-Vey class associated to a codimension one foliation; see e.g. Candel-Conlon volume 2, Chapter 7). Identifying the universal cover of each leaf with $\widetilde{M}$ by projection, the fact that our measure is harmonic means that $\beta$ “is” the gradient of the logarithm of a positive harmonic function on $\widetilde{M}$. As observed by Thurston, the geometry of $M$ then puts constraints on the size of $\beta$. The following discussion is taken largely from Thurston’s paper “Three-manifolds, foliations and circles II” (unfortunately this mostly unwritten paper is not publicly available; some details can be found in my foliations book, example 4.6).

An orthogonal connection on $E$ can be obtained by averaging $\alpha$ under the flow of $X$; i.e. if $\phi_t$ is the diffeomorphism of $E$ which rotates each circle through angle $t$, then

$\omega = \frac {1} {2\pi} \int_{0}^{2\pi} \phi_t^* \alpha$

is an $X$-invariant $1$-form on $E$, which therefore descends to a $1$-form on $M$, which can be thought of as a connection form for an $\text{SO}(2)$-structure on the bundle $E$. The curvature of the connection (in the usual sense) is the $2$-form $d\omega$, and we have a formula

$d\omega = \frac {1} {2\pi} \int_{0}^{2\pi} \phi_t^*(d\alpha) = \frac {1} {2\pi} \int_{0}^{2\pi} \phi_t^*(-\beta \wedge \alpha)$

The action of the $1$-parameter group $\phi_t$ trivializes the cotangent bundle to $E$ over each fiber. After choosing such a trivialization, we can think of the values of $\alpha$ at each point on a fiber as sweeping out a circle $\gamma$ in a fixed vector space $V$. The tangent to this circle is found by taking the Lie derivative

$\mathcal{L}_X(\alpha) = \iota_X d\alpha + d\iota_X \alpha = \alpha(X)\beta = \beta$

In other words, $\beta$ is identified with $d\gamma$ under the identification of $\alpha$ with $\gamma$, and $\int \phi_t^*(-\beta \wedge \alpha) = \int \gamma \wedge d\gamma$; i.e. the absolute value of the curvature of the connection is equal to $1/\pi$ times the area enclosed by $\gamma$.

Now suppose $M$ is a hyperbolic $n$-manifold, i.e. a manifold of dimension $n$ with constant curvature $-1$ everywhere. Equivalently, think of $M$ as a quotient of hyperbolic space $\mathbb{H}^n$ by a discrete group of isometries. A positive harmonic function on $\mathbb{H}^n$ has a logarithmic derivative which is bounded pointwise by $(n-1)$; identifying positive harmonic functions on hyperbolic space with distributions on the sphere at infinity, one sees that the  “worst case” is the harmonic extension of an atomic measure concentrated at a single point at infinity, since every other positive harmonic function is the weighted average of such examples. As one moves towards or away from a blob at infinity concentrated near this point, the radius of the blob expands like $e^t$; since the sphere at infinity has dimension $n-1$, the conclusion follows. But this means that the speed of $\gamma$ (i.e. the size of $d\gamma$) is pointwise bounded by $(n-1)$, and the length of the $\gamma$ circle is at most $2\pi(n-1)$. A circle of length $2\pi(n-1)$ can enclose a disk of area at most $\pi (n-1)^2$, so the curvature of the connection has absolute value pointwise bounded by $(n-1)^2$.

One corollary is a new proof of the Milnor-Wood inequality, which says that a foliated circle bundle $E$ over a closed oriented surface $S$ of genus at least $2$ satisfies $|e(E)| \le -\chi(S)$, where $e(E)$ is the Euler number of the bundle (a topological invariant). For, the surface $S$ can be given a hyperbolic metric, and the bundle a harmonic connection whose average is an orthogonal connection with pointwise curvature of absolute value at most $1$. The Euler class of the bundle evaluated on the fundamental class of $S$ is the Euler number $e(E)$; we have

$|e(E)| = \frac {1} {2\pi} |\int_S \omega| \le \text{area}(S)/2\pi = -\chi(S)$

where the first equality is the Chern-Weil formula for the Euler class of a bundle in terms of the curvature of a connection, and the last equality is the Gauss-Bonnet theorem for a hyperbolic surface. Another corollary gives lower bounds on the area of an incompressible surface in a hyperbolic manifold. Suppose $S \to M$ is an immersion which is injective on $\pi_1$. There is a cover $\widehat{M}$ of $M$ for which the immersion lifts to a homotopy equivalence, and we get an action of $\pi_1(\widehat{M})$ on the circle at infinity of $S$, and hence a foliated circle bundle as above with $e(E) = -\chi(S)$. Integrating as above over the image of $S$ in $\widehat{M}$, and using the fact that the curvature of $\omega$ is pointwise bounded by $(n-1)^2$, we deduce that the area of $S$ is at least $-2\pi\chi(S)/(n-1)^2$. If $M$ is a $3$-manifold, we obtain $\text{area}(S) \ge -2\pi\chi(S)/4$.

(A somewhat more subtle argument allows one to get better bounds, e.g. replacing $4$ by $(\pi/2)^2$ for $n=3$, and better estimates for higher $n$.)