In this note, we prove that a random extension of either the free group FN of rank N ě 3 or of the fundamental group of a closed, orientable surface Sg of genus g ě 2 is a hyperbolic group. Here, a random extension is one corresponding to a subgroup of either OutpFN q or ModpSgq generated by k independent random walks. Our main theorem is that a k–generated random subgroup of ModpSgq or OutpFN ...

A plane is a two-dimensional right vector space V over a division algebra D, which we assume in this paper has an involution; a hermitian form h : V × V −→ D is hyperbolic, if, with respect to a properly choosen basis, it is given by the matrix 0 1 1 0 ) ; the pair (V, h) is called a hyperbolic plane. If D is central simple over a number field K, a hyperbolic plane (V, h) gives rise to a reduct...

Contents 1. Hyperbolic geometry 1 2. Fuchsian groups and hyperbolic surfaces 4 3. Spectrum and resolvent 11 4. Spectral theory: finite-area case 16 5. Spectral theory: infinite-area case 20 6. Selberg trace formula 23 7. Arithmetic surfaces 33 References 44 1. Hyperbolic geometry In complex analysis, we learn that the upper half-plane H = {Im z > 0} has a large group of conformal automorphisms,...

We introduce the co-surface graph CS of a finitely generated free group F and use it to study the geometry of hyperbolic group extensions of F. Among other things, we show that the Gromov boundary of the co-surface graph is equivariantly homeomorphic to the space of free arational F–trees and use this to prove that a finitely generated subgroup of Out(F) quasi-isometrically embeds into the co-s...

A comparison between Tsallis' statistics-based and generalized quasi-hyperbolic discount models in humans. Acknowledgements: The research reported in this paper was supported by a grant from the Grant-in-Aid for Scientific Research (" 21st century center of excellence " grant) from the Ministry of Education, Culture, Sports, Science and Technology of Japan, and Yamaguchi endocrinological disord...

In the present paper, we introduce and discuss the notion of monodromically full points of configuration spaces of hyperbolic curves. This notion leads to complements to M. Matsumoto’s result concerning the difference between the kernels of the natural homomorphisms associated to a hyperbolic curve and its point from the Galois group to the automorphism and outer automorphism groups of the geom...

Let G be a word hyperbolic group in the sense of Gromov and P its associated Rips complex. We prove that the fixed point set PH is contractible for every finite subgroup H of G. This is the main ingredient for proving that P is a finite model for the universal space EG for proper actions. As a corollary we get that a hyperbolic group has only finitely many conjugacy classes of finite subgroups.

in this paper, we prove that every metric line in the poincare ball model of hyperbolic geometry is exactly a classical line of itself. we also proved nonexistence of periodic lines in the poincare ball model of hyperbolic geometry.

Given a finitely generated subgroup Γ≤ Out(F) of the outer automorphism group of the rank r free group F = Fr, there is a corresponding free group extension 1→ F→ EΓ→ Γ→ 1. We give sufficient conditions for when the extension EΓ is hyperbolic. In particular, we show that if all infinite order elements of Γ are atoroidal and the action of Γ on the free factor complex of F has a quasi-isometric o...

In this paper, we provide a complete theory of Diophantine approximation in the limit set of a group acting on a Gromov hyperbolic metric space. This summarizes and completes what has until now been an ad hoc collection of results by many authors. In addition to providing much greater generality than any prior work, our results also give new insight into the nature of the connection between Dio...