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Results tagged with hyperbolic-geometry
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user 13825
13
votes
2
answers
984
views
Measure on the Boundary of a Hyperbolic Group
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 …
- 3,626
13
votes
1
answer
948
views
Gromov's Hyperbolicity and Positive Cheeger Constant in Planar Graphs
It is known that for metric graphs the concepts of Gromov's hyperbolicity and strictly positive Cheeger constant are related. Let us first recall the definition of the Cheeger constant. Let $G$ be a g …
- 3,626
4
votes
2
answers
669
views
Capacity of Balls in Hyperbolic Space
Given $M$ a Riemannian manifold and $\Omega\subset M$ the capacity of $\Omega$ is defined as
$$
\mathrm{cap}(\Omega)=\inf \int_{M\setminus\Omega}{|\mathrm{grad} \varphi|^2 dV}
$$
where $\varphi$ range …
- 3,626
10
votes
3
answers
2k
views
Hyperbolicity on Riemann Surfaces
For Riemann surfaces there are at least to possible notions of hyperbolicity. The classical one given by the Uniformization Theorem, or equivalently the type problem, which essentially says that a sim …
- 3,626