Arithmetic of hyperbolic 3-manifolds

By I. Nikolaev. 2002

437 citations recorded by the source corpus.

This note is an elaboration of the ideas and intuitions of Grothendieck and Weil concerning the "arithmetic topology". Given 3-dimensional manifold M fibering over the circle we introduce an real quadratic number field K with discriminant d, where d>0 is an integer number uniquely determined by M. The idea is to relate geometry of M to the arithmetic of field K. On this way, we show that Vol M is a limit density of ideals of given norm in the field K (Dirichlet density). The second statement says that the number of cusp points of manifold M is equal to the class number of the field K. It is remarkable that both of the invariants can be explicitly calculated for the concrete values of discriminant d. Our approach is based on the K-theory of noncommutative C*-algebras coming from measured foliations and geodesic laminations studied by Thurston et al. We apply the elaborated technique to solve the Poincare conjecture for given class of manifolds.

View Paper: https://www.semanticscholar.org/paper/5659931cba153ea86b65f929078504a73bde0811

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

Citations in the corpus

Geometric Group Theory (2018, methodology): “We refer the reader to [MR03] for the detailed discussion of these and other facts about Bianchi groups.”
A torsion Jacquet--Langlands correspondence (2012, methodology): “We refer to [53] for background and further details on these types of construction, especially Chapter 6 for orders in quaternion algebras.”
Compactifications of Symmetric and Locally Symmetric Spaces (2005, methodology): “There are also co-compact arithmetic subgroups of SL(2,C) constructed via quaternion algebras over F (see [MR]).”
Arithmetic Fuchsian Groups of Genus Zero (2006, methodology): “We summarize for convenience the classification theorem for quaternion algebras over number fields (see [27] Chapter 7 or [44] Chapter 3).”
Geodesics and commensurability classes of arithmetic hyperbolic $3$-manifolds (2008, background): “In this section we recall some facts about arithmetic Kleinian groups Γ ⊂ PSL2(C); see [8] for details.”
Counting arithmetic lattices and surfaces (2008, methodology): “An element of m(A) is denoted by ΓS,D where D is a maximal order in A (in [MR] it is denoted by O but we prefer to use D as O for us is the ring of integers in k) and S is a finite set of finite primes of k disjoint from Ramf (A).”
Flexing closed hyperbolic manifolds. (2007, background): “5 of [13] that the manifold vol3 contains no immersed, closed, totally geodesic surface (indeed its fundamental group contains no nonelementary Fuchsian subgroup); moreover, N.”
Absolute profinite rigidity and hyperbolic geometry (2018, background): “By class field theory, a quaternion algebra B/K is determined up to K–isomorphism by the set of Kv–isomorphism classes {Bv}v∈VK ; in particular, B is determined by Ram(B), which is finite and of even cardinality (see for example [32]).”
LOGARITHMIC GROWTH OF SYSTOLE OF ARITHMETIC RIEMANN SURFACES ALONG CONGRUENCE SUBGROUPS (2005, background): “For additional background on quaternion algebras and arithmetic Fuchsian groups, see [GP69, Appendix to chapter 1], [K92, Section 5.2] and [ MR03 ].”
Cohomological goodness and the profinite completion of Bianchi groups (2008, background): “For a detailed description and the basic properties of them see [5, Section 7] and [17].”
The Ergodic Theory of Hyperbolic Groups (2011, background): “It is easy to produce examples for which the groups in question are not commensurable; for instance, a pair of closed hyperbolic 3-manifolds M1, M2 with different invariant trace fields (see [27]).”
Prescribing the behaviour of geodesics in negative curvature (2007, unknown): “By varying the (nonuniform) arithmetic lattices in the isometry group of a negatively curved symmetric space (see for instance Maclachlan and Reid [42] or Maclachlan, Waterman and Wielenberg [43]), other arithmetic applications are possible.”