Chebyshev polynomials and inequalities for Kleinian groups

The principal character of a representation of the free group of rank two into [Formula: see text] is a triple of complex numbers that determines an irreducible representation uniquely up to conjugacy. It is a central problem in the geometry of discrete groups and low dimensional topology to determine when such a triple represents a discrete group which is not virtually abelian, that is, a Kleinian group. A classical necessary condition is Jørgensen’s inequality. Here, we use certain shifted Chebyshev polynomials and trace identities to determine new families of such inequalities, some of which are best possible. The use of these polynomials also shows how we can identify the principal character of some important subgroups from that of the group itself.

Geometry of Kleinian groups and its applications

The notion of Kleinian groups was first introduced by Poincaré in the 19th century, and their study from a viewpoint of complex analysis was developed by Ahlfors, Bers, Kra, Maskit and Marden among others. After Thurston’s innovative work, topological study of Kleinian groups was started and flourished. In this survey article, we look back on Thurston’s influential work and his famous 10 open questions, and explain their significance and how they have been solved. In the latter half, we shall look at more recent development of the theory of Kleinian groups.

Horospherically invariant measures and finitely generated Kleinian groups

<p style='text-indent:20px;'>Let <inline-formula><tex-math id="M1">\begin{document}$ \Gamma &lt; {\rm{PSL}}_2( \mathbb{C}) $\end{document}</tex-math></inline-formula> be a Zariski dense finitely generated Kleinian group. We show all Radon measures on <inline-formula><tex-math id="M2">\begin{document}$ {\rm{PSL}}_2( \mathbb{C}) / \Gamma $\end{document}</tex-math></inline-formula> which are ergodic and invariant under the action of the horospherical subgroup are either supported on a single closed horospherical orbit or quasi-invariant with respect to the geodesic frame flow and its centralizer. We do this by applying a result of Landesberg and Lindenstrauss [<xref ref-type="bibr" rid="b18">18</xref>] together with fundamental results in the theory of 3-manifolds, most notably the Tameness Theorem by Agol [<xref ref-type="bibr" rid="b2">2</xref>] and Calegari-Gabai [<xref ref-type="bibr" rid="b10">10</xref>].</p>

Distribution of Modular Symbols in ℍ3

We introduce a new technique for the study of the distribution of modular symbols, which we apply to the congruence subgroups of Bianchi groups. We prove that if $K$ is a quadratic imaginary number field of class number one and $\mathcal{O}_K$ is its ring of integers, then for certain congruence subgroups of $\textrm{PSL}_2(\mathcal{O}_K)$, the periods of a cusp form of weight two obey asymptotically a normal distribution. These results are specialisations from the more general setting of quotient surfaces of cofinite Kleinian groups where our methods apply. We avoid the method of moments. Our new insight is to use the behaviour of the smallest eigenvalue of the Laplacian for spaces twisted by modular symbols. Our approach also recovers the first and second moments of the distribution.