Hyperbolic Geometry from a Local Viewpoint

The Nielsen-Thurston Classification

10.23943/princeton/9780691147949.003.0014 ◽

2017 ◽

Author(s):

Benson Farb ◽

Dan Margalit

Keyword(s):

Hyperbolic Geometry ◽

Mapping Class Groups ◽

Classification Theorem ◽

Mapping Class ◽

Fundamental Result ◽

Class Groups ◽

Mapping Torus ◽

Higher Genus ◽

Manifold Theory

This chapter explains and proves the Nielsen–Thurston classification of elements of Mod(S), one of the central theorems in the study of mapping class groups. It first considers the classification of elements for the torus of Mod(T² before discussing higher-genus analogues for each of the three types of elements of Mod(T². It then states the Nielsen–Thurston classification theorem in various forms, as well as a connection to 3-manifold theory, along with Thurston's geometric classification of mapping torus. The rest of the chapter is devoted to Bers' proof of the Nielsen–Thurston classification. The collar lemma is highlighted as a new ingredient, as it is also a fundamental result in the hyperbolic geometry of surfaces.

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Counting and equidistribution in quaternionic Heisenberg groups

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004121000426 ◽

2021 ◽

pp. 1-38

Author(s):

JOUNI PARKKONEN ◽

FRÉDÉRIC PAULIN

Keyword(s):

Hyperbolic Geometry ◽

Quaternion Algebra ◽

Rational Points ◽

Hyperbolic Spaces ◽

Heisenberg Groups ◽

Arithmetic Groups ◽

Hermitian Forms ◽

Dimension 2 ◽

Counting Formula ◽

The Relationship

Abstract We develop the relationship between quaternionic hyperbolic geometry and arithmetic counting or equidistribution applications, that arises from the action of arithmetic groups on quaternionic hyperbolic spaces, especially in dimension 2. We prove a Mertens counting formula for the rational points over a definite quaternion algebra A over ${\mathbb{Q}}$ in the light cone of quaternionic Hermitian forms, as well as a Neville equidistribution theorem of the set of rational points over A in quaternionic Heisenberg groups.

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