I consider p-Bernoulli bond percolation on transitive, nonamenable, planar graphs with one end and on their duals. It is known from [BS01] that in such a graph G we have three essential phases of percolation, i.e. $0<p_c\left(G\right)<p_u\left(G\right)<1$, where $p_c$ is the critical probability and $p_u$-the unification probability. I prove that in the middle phase a.s. all the ends of all the infinite clusters have one-point boundaries in ∂ℍ². This result is similar to some results in [Lal].

Hecke groups $H\left({\lambda }_q\right)$ are the discrete subgroups of $\mathrm{P}SL(2,ℝ)$ generated by $S\left(z\right)=-{(z+{\lambda }_q)}^{-1}$ and $T\left(z\right)=-\frac{1}{z}$. The commutator subgroup of $H$(${\lambda }_q)$, denoted by $H^{\text{'}}\left({\lambda }_q\right)$, is studied in [2]. It was shown that $H^{\text{'}}\left({\lambda }_q\right)$ is a free group of rank $q-1$. Here the extended Hecke groups $\overline{H}\left({\lambda }_q\right)$, obtained by adjoining $R_1\left(z\right)=1/\overline{z}$ to the generators of $H\left({\lambda }_q\right)$, are considered. The commutator subgroup of $\overline{H}\left({\lambda }_q\right)$ is shown to be a free product of two finite cyclic groups. Also it is interesting to note that while in the $H\left({\lambda }_q\right)$ case, the index of $H^{\text{'}}\left({\lambda }_q\right)$ is changed by $q$, in the case of $\overline{H}\left({\lambda }_q\right)$, this number is either 4 for...

We exhibit an algorithm to compute a Dirichlet domain for a Fuchsian group $\Gamma$ with cofinite area. As a consequence, we compute the invariants of $\Gamma$, including an explicit finite presentation for $\Gamma$.

We study two classes of linear representations of a surface group: Hitchin and maximal symplectic representations. We relate them to cross ratios and thus deduce that they are displacing which means that their translation lengths are roughly controlled by the translations lengths on the Cayley graph. As a consequence, we show that the mapping class group acts properly on the space of representations and that the energy functional associated to such a representation is proper. This implies the existence...

This article relates representations of surface groups to cross ratios. We first identify a connected component of the space of representations into PSL(n,ℝ) – known as the n-Hitchin component– to a subset of the set of cross ratios on the boundary at infinity of the group. Similarly, we study some representations into $C^{1,h}\left(𝕋\right)⋊{\text{Diff}}^h\left(𝕋\right)$ associated to cross ratios and exhibit a “character variety” of these representations. We show that this character variety contains alln-Hitchin components as well as the set of...