Showing papers on "Arithmetic group published in 2007"
TL;DR: In this article, a trace estimate valid for an arbitrary ideal in a quaternion algebra was obtained for a principal congruence tower of Hurwitz surfaces (PCH) and a 4/3 bound for the systole of hyperbolic 3-manifolds, relative to their simplicial volume.
Abstract: We apply a study of orders in quaternion algebras, to the differential geometry of Riemann surfaces. The least length of a closed geodesic on a hyperbolic surface is called its systole, and denoted sysπ 1 . P. Buser and P. Sarnak constructed Riemann surfaces X whose systole behaves logarithmically in the genus g(X). The Fuchsian groups in their examples are principal congruence subgroups of a fixed arithmetic group with rational trace field. We generalize their construction to principal congruence subgroups of arbitrary arithmetic surfaces. The key tool is a new trace estimate valid for an arbitrary ideal in a quaternion algebra. We obtain a particularly sharp bound for a principal congruence tower of Hurwitz surfaces (PCH), namely the 4/3bound sysπ1(XPCH) ≥ 4 log(g(XPCH)). Similar results are obtained for the systole of hyperbolic 3-manifolds, relative to their simplicial volume.
95 citations
TL;DR: In this paper, the residual properties of finitely generated linear groups were studied and it was shown that each link is a sublink of some link whose group is residually 2-finite.
Abstract: We study the residual properties of finitely generated linear groups. Using the methods under consideration, we prove the residual 2-finiteness of the groups of the Whitehead link, the Borromean links (answering a question of Cochran), and some other links. We show also that each link is a sublink of some link whose group is residually 2-finite.
8 citations