Measure on the Boundary of a Hyperbolic Group - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T23:54:50Z http://mathoverflow.net/feeds/question/90531 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/90531/measure-on-the-boundary-of-a-hyperbolic-group Measure on the Boundary of a Hyperbolic Group ght 2012-03-08T01:57:55Z 2012-03-10T03:28:09Z <p>Let $\Gamma$ be a non-elementary Gromov's $\delta$-hyperbolic group. Let $B(1,n)$ be the set of elements at distance at most $n$ from the identity and let $\partial B(1,n)$ be the elements at distance $n$ (with the word metric). Consider the probability measures $\mu_n$ and $\nu_n$ defined as $$\mu_n := \frac{1}{|B(1,n)|}\sum_{\gamma\, \in B(1,n)}{\delta_{\gamma}}$$ and $$\nu_n := \frac{1}{|\partial B(1,n)|}\sum_{\gamma\,\in\partial B(1,n)}{\delta_{\gamma}}.$$ It is clear that these measures converge weakly (there is a convergent sub-sequence) on the compact space $\Gamma\cup\partial\Gamma$. Moreover, the limit measure is supported on $\partial\Gamma$.</p> <p>My question are:</p> <ul> <li>Does anybody have study the limit measures?</li> <li>Are these related with the Hausdorff measure on the boundary? to the Patterson-Sullivan measure? to the harmonic measure?</li> </ul> http://mathoverflow.net/questions/90531/measure-on-the-boundary-of-a-hyperbolic-group/90533#90533 Answer by Lee Mosher for Measure on the Boundary of a Hyperbolic Group Lee Mosher 2012-03-08T02:40:15Z 2012-03-08T02:40:15Z <p>I don't know about the specific sums you suggest, but here are some well established alternatives.</p> <p>Try Kaimanovich's paper "The Poisson boundary of hyperbolic groups", which is about boundaries arising from random walks, which are shown to coincide with the Gromov boundary.</p> <p>Another somewhat different approach is found in Coornaert's paper <a href="http://projecteuclid.org/DPubS?service=UI&amp;version=1.0&amp;verb=Display&amp;handle=euclid.pjm/1102634263" rel="nofollow">Patterson-Sullivan measures on the boundary of a hyperbolic space in the sense of Gromov</a>. The idea, which goes back to the original Patterson-Sullivan measures on limit sets of Kleinian groups, is to construct measures on the boundary using sums over the whole group, where group elements are weighted by a function that decays exponentially in the distance; the base of the exponential is chosen delicately, based on geometric properties of the group. The resulting measures on the boundary are quasiconformal'' measures. </p>