Vega Peleteiro, Javier (2026) The Geometry of Free Groups and the Ping-Pong Lemma. Bachelor's Thesis, Mathematics.
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Abstract
The ping-pong lemma is a tool in geometric group theory used to find free subgroups inside larger groups, by letting them act on specific sets. This paper explores different applications and generalizations of the ping-pong lemma across several branches of mathematics. We begin by stating the classical version of the lemma and its proof, as well as a demonstration of the mechanism in GL(2,Z). We subsequently explore its application to automorphisms of trees (Aut(T)), automorphisms of the Riemann sphere (PSL(2,C)), isometries of the hyperbolic plane (PSL(2,R)), and the mapping class group (Mod(S)). We then provide a generalization to establish free products, proving that PSL(2,Z) ≅ Z/2Z ∗ Z/3Z. Finally, we adapt the ping-pong lemma to more complex groups with certain nontrivial relations: we extend the lemma to construct free groups with partial commutation relations (right-angled Artin groups), and we finally formulate a version to determine amalgamated free products, and use it to prove the classical result SL(2,Z) ≅ Z/4Z ∗(Z/2Z) Z/6Z.
| Item Type: | Thesis (Bachelor's Thesis) |
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| Supervisor name: | Karakurt, C. and Veen, R.I. van der |
| Degree programme: | Mathematics |
| Thesis type: | Bachelor's Thesis |
| Language: | English |
| Date Deposited: | 10 Jul 2026 14:32 |
| Last Modified: | 10 Jul 2026 14:32 |
| URI: | https://fse.studenttheses.ub.rug.nl/id/eprint/37416 |
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