A note on the Burnside problem for homeomorphism groups of manifolds

Abstract: This note studies the Burnside problem for homeomorphism groups of compact connected manifolds. For surfaces, we prove that the identity component of the homeomorphism group is torsion-free precisely when the surface is not the sphere, torus, projective plane, or Klein bottle. An extension argument based on the Tits alternative for mapping class groups then implies that every finitely generated periodic subgroup of the full homeomorphism group is finite for all surfaces outside this exceptional list, recovering and extending a theorem of Guelman and Liousse to non-orientable surfaces. For the circle, we prove that every finitely generated periodic subgroup of its homeomorphism group is finite and cyclic. We close with remarks on manifolds with boundary and open questions on the Burnside problem for hyperbolic three-manifolds and doubled handlebodies.

• The paper provides a complete characterization of when the identity component of surface homeomorphism groups is torsion-free, identifying specific exceptional surfaces.
• It employs topological, braid group, and mapping class group techniques to extend the Burnside problem framework to both orientable and non-orientable cases.
• The analysis also covers periodic subgroup behaviors in circle and interval homeomorphisms, and outlines open questions for higher-dimensional manifolds.

The Burnside Problem for Homeomorphism Groups of Manifolds

Introduction

This paper investigates the Burnside problem in the context of homeomorphism groups of compact, connected manifolds, with a primary emphasis on surfaces. The Burnside problem asks whether every finitely generated periodic group is necessarily finite. While algebraic counterexamples are known, the problem's status for various topological transformation groups remains a central line of inquiry. The paper provides a comprehensive treatment for surfaces, establishes new results for non-orientable cases, addresses the structure of periodic subgroups in homeomorphism groups of the circle and the interval, and discusses the implications and open directions for 3-manifolds.

Main Results: Surface Homeomorphism Groups

The principal result is a complete characterization of surfaces $\Sigma$ for which the identity component $\Homeo_0(\Sigma)$ of the homeomorphism group is torsion-free. The central theorem asserts that $\Homeo_0(\Sigma)$ is torsion-free if and only if $\Sigma$ is not homeomorphic to the sphere, torus, projective plane, or Klein bottle. This equivalence is established via:

• Topological arguments leveraging homotopy and braid group techniques.
• Explicit construction of nontrivial torsion elements in $\Homeo_0(\Sigma)$ for each exceptional case ($\mathbb{S}^2, T^2, \mathbb{R}P^2, K$).
• A proof that for all other surfaces, braid group and mapping class group structure forces torsion-freeness.

The result recovers, generalizes, and provides new proofs of previously known theorems, notably extending Guelman-Liousse's analysis to non-orientable surfaces (2604.15627).

The Burnside Property Beyond the Identity Component

Utilizing the Tits alternative for mapping class groups—namely, that all finitely generated subgroups are either virtually solvable or contain a non-abelian free group—the analysis extends to the full homeomorphism group. The argument proceeds via:

• Proving that any finite extension of groups with the Burnside property also has the Burnside property, provided the kernel is torsion-free.
• Noting that mapping class groups satisfy the Tits alternative in both orientable and non-orientable cases.
• Deducing that for all surfaces $$\Sigma \notin \{\mathbb{S}^2, T^2, \mathbb{R}P^2, K\}$$, every finitely generated periodic subgroup of $\Homeo(\Sigma)$ is finite.

This extension principle is formalized and precisely justified, reinforcing the structural link between homeomorphism groups and their associated mapping class groups (2604.15627).

The Exceptional Surfaces and Open Problems

For $\mathbb{S}^2, T^2, \mathbb{R}P^2$, and $K$, the Burnside problem remains unresolved for the topological homeomorphism groups due to the existence of torsion in the identity component. The known partial results include:

• Finiteness of such subgroups under analytic or measure-preserving constraints (e.g., area-preserving analytic diffeomorphisms of $\Homeo_0(\Sigma)$0 [MR4173155], measure-preserving homeomorphisms of $\Homeo_0(\Sigma)$1 [MR3158049]), but not for topological groups.
• Some bounded exponent results (e.g., powers of $\Homeo_0(\Sigma)$2 for $\Homeo_0(\Sigma)$3 [MR3892240]).
• The literature contains no established results for the projective plane or Klein bottle for the full homeomorphism group.

Therefore, the Burnside property is conjectured for these four surfaces but requires development of new methods beyond those established for other surfaces.

Periodic Subgroups for Circle and Interval Homeomorphism Groups

A complete solution for the Burnside problem is provided for the circle and interval:

• Every finitely generated periodic subgroup of $\Homeo_0(\Sigma)$4 is trivial, by virtue of torsion-freeness ([MR3560661], (2604.15627)).
• Every finitely generated periodic subgroup of $\Homeo_0(\Sigma)$5 is finite and cyclic. The proof utilizes fixed point theory and the classification of finite subgroups of orientation-preserving circle homeomorphisms, relying on the faithful action of such groups on finite orbits and reduction to cyclicity ([MR2809110], [MR3813208]).

Extending to Higher Dimensions: Manifolds with Boundary and Closed 3-Manifolds

For manifolds with boundary, an immediate consequence of Newman's theorem and subsequent refinements is that $\Homeo_0(\Sigma)$6 is torsion-free: any nontrivial periodic homeomorphism must move points in the interior ([N31], [MR4128], [MR238353], [MR413144]). Therefore, the Burnside property is immediate.

For closed 3-manifolds, especially closed hyperbolic ones, the mapping class group is finite by Mostow rigidity. The Burnside problem then reduces to its status for the identity component, but this remains open. Similar questions are proposed for doubled handlebodies, where mapping class group structure suggests the Tits alternative holds, but the analysis of $\Homeo_0(\Sigma)$7 is yet unsettled.

Implications and Further Directions

The paper illuminates a deep interplay between topology, geometric group theory, and dynamical systems within the context of transformation groups. By recasting the Burnside problem in analytic, measure-theoretic, and topological settings, and deploying braid group and mapping class group techniques, the work provides a unified and systematic criteria for torsion-freeness and the Burnside property among surface homeomorphism groups, both orientable and non-orientable.

The methods suggest new lines of attack for the Burnside property in higher dimensions, especially in links with Mostow rigidity and Out$\Homeo_0(\Sigma)$8. The unresolved cases for certain surfaces invite theoretical advances possibly through dynamical, ergodic, or cohomological techniques. The treatment of periodic subgroups in the circle and interval context further illustrates the delicate relationship between group actions, topology, and global symmetries.

Conclusion

This paper establishes a thorough and structured understanding of the Burnside problem for homeomorphism groups of surfaces, generalizing earlier orientable results to non-orientable cases and providing new, unified proofs grounded in geometric group theory. Several open cases and questions persist, most notably for the four exceptional surfaces and some 3-manifolds, laying the groundwork for future exploration into the torsion structure and periodic group actions in topological transformation groups (2604.15627).