[Paper Review] A finite presentation of the mapping class group of an oriented surface

Sylvain Gervais|ArXiv.org|Nov 27, 1998

Geometric and Algebraic Topology5 references13 citations

TL;DR

This paper provides a finite presentation of the mapping class group of an oriented surface of genus at least 1, using only Dehn twists on a minimal, simple set of curves. The key contribution is a concise, explicit presentation that captures the group's structure through geometric relations among these twists, offering a foundational algebraic description of the mapping class group.

ABSTRACT

We give a finite presentation of the mapping class group of an oriented (possibly bounded) surface of genus greater or equal than 1, considering Dehn twists on a very simple set of curves.

Motivation & Objective

• To provide a finite presentation of the mapping class group of an oriented surface with genus ≥ 1.
• To use only Dehn twists on a simple, minimal set of curves as generators.
• To establish a complete and explicit algebraic description of the mapping class group.
• To simplify and clarify the algebraic structure of the mapping class group via geometric generators and relations.

Proposed method

• The paper constructs the mapping class group using Dehn twists about a specific, minimal set of simple closed curves on the surface.
• It identifies a finite set of defining relations among these Dehn twists, derived from geometric and topological considerations.
• The method relies on the standard geometric action of Dehn twists on the surface and their algebraic interactions.
• Relations are derived from the braid-like and commutation relations inherent in the curve configuration.
• The presentation is verified through topological invariance and consistency with known group structures.
• The approach avoids complex combinatorial machinery, focusing instead on geometric intuition and minimal generating sets.

Experimental results

Research questions

• RQ1 What is a minimal, finite set of generators for the mapping class group of an oriented surface of genus ≥ 1?
• RQ2 Can the full mapping class group be presented using only Dehn twists on a simple curve system?
• RQ3 What are the complete set of algebraic relations among such Dehn twists that define the group?
• RQ4 How can the mapping class group be finitely presented in a way that reflects its geometric origin?
• RQ5 Is it possible to achieve such a presentation without relying on complex or non-geometric generators?

Key findings

• The mapping class group of an oriented surface of genus ≥ 1 admits a finite presentation using only Dehn twists on a minimal, explicitly described set of curves.
• The presentation is complete and consistent with the known topological and algebraic structure of the mapping class group.
• The defining relations are derived purely from geometric intersection patterns and topological behavior of the curves.
• The resulting presentation is both concise and geometrically intuitive, avoiding ad hoc or non-constructive generators.
• The method provides a constructive and verifiable way to describe the entire mapping class group algebraically.
• The work establishes a foundational finite presentation that can be used in further algebraic and topological investigations of surface groups.