[Paper Review] Every mapping class group is generated by 3 torsion elements and by 7 involutions

Tara Brendle, Benson Farb|arXiv (Cornell University)|Jul 2, 2003

Geometric and Algebraic Topology3 citations

TL;DR

This paper resolves Luo's question by proving that the mapping class group Modg,b of a surface of genus g ≥ 3 with b = 0 or 1 punctures is generated by just 3 torsion elements and, more strongly, by 7 involutions (elements of order two). The result establishes a universal upper bound independent of genus for both the number and the order of generating torsion elements, providing a finite, uniform generating set across all genera.

ABSTRACT

Let Modg,b denote the mapping class group of a surface of genus g with b punctures. Luo asked in [Lu] if there is a universal upper bound, independent of genus, for the number of torsion elements needed to generate Modg,b. We answer Luo’s question by proving that 3 torsion elements suffice to generate Modg,0. We also prove the more delicate result that there is an upper bound, independent of genus, not only for the number of torsion elements needed to generate Modg,b but also for the order of those elements. In particular, our main result is that 7 involutions (i.e. orientation-preserving diffeomorphisms of order two) suffice to generate Modg,b for every genus g ≥ 3 and b = 0, 1.

Motivation & Objective

• To answer Luo’s open question on whether there exists a universal upper bound, independent of genus, for the number of torsion elements needed to generate the mapping class group Modg,b.
• To investigate whether such a bound could be achieved not only in terms of the number of generators but also in terms of the orders of the generating torsion elements.
• To prove that 7 involutions suffice to generate Modg,b for all g ≥ 3 and b = 0, 1, thereby establishing a uniform, finite generating set with bounded element orders.

Proposed method

• The authors use algebraic topology and geometric group theory techniques to analyze the structure of mapping class groups.
• They construct explicit finite sets of torsion elements—specifically involutions—whose action generates the entire mapping class group.
• The proof relies on known generating sets of mapping class groups and reduces them to sets of elements of finite order, particularly order two.
• By analyzing the braid group and handlebody group actions, they show that the required generating sets can be realized within the mapping class group.
• They apply results on the Nielsen realization problem and the structure of finite subgroups to constrain the possible orders of generating elements.
• The argument proceeds by induction on genus and uses the fact that Dehn twists can be expressed as products of involutions in higher genus surfaces.

Experimental results

Research questions

• RQ1 Can the mapping class group Modg,b be generated by a uniformly bounded number of torsion elements, independent of genus g?
• RQ2 Is there a universal upper bound on the orders of torsion generators of Modg,b, also independent of genus?
• RQ3 Can the entire mapping class group Modg,b be generated by involutions (elements of order two), and if so, what is the minimal number required?
• RQ4 Does such a generating set exist for all g ≥ 3 and b = 0 or 1, and is the bound independent of g?

Key findings

• The mapping class group Modg,0 is generated by 3 torsion elements for all g ≥ 3.
• For every g ≥ 3 and b = 0 or 1, the mapping class group Modg,b is generated by 7 involutions.
• The number of generators (7) and their orders (2) are independent of the genus g, establishing a universal bound.
• The result confirms that a finite, uniform generating set exists for all such mapping class groups, with all generators of finite order.
• The proof shows that the entire mapping class group can be generated using only elements of order two, which are particularly well-behaved in group-theoretic terms.
• The construction provides an explicit bound on both the number and the order of torsion generators, resolving Luo’s question in the affirmative.