Isometry groups of infinite-genus hyperbolic surfaces (2007.01982v2)

Abstract: Given a 2-manifold, a fundamental question to ask is which groups can be realized as the isometry group of a Riemannan metric of constant curvature on the manifold. In this paper, we give a nearly complete classification of such groups for infinite-genus 2-manifolds with no planar ends. Surprisingly, we show there is an uncountable class of such 2-manifolds where every countable group can be realized as an isometry group (namely, those with self-similar end spaces). We apply this result to obtain obstructions to standard group theoretic properties for the groups of homeomorphisms, diffeomorphisms, and the mapping class groups of such 2-manifolds. For example, none of these groups satisfy the Tits Alternative; are coherent; are linear; are cyclically or linearly orderable; or are residually finite. As a second application, we give an algebraic rigidity result for mapping class groups.

• The paper shows that every countable group can be realized as the isometry group on infinite-genus hyperbolic surfaces with self-similar ends.
• It classifies surfaces based on end space structure, revealing that doubly pointed and non-displaceable ends yield restricted symmetries.
• It analyzes algebraic properties, highlighting deviations from the Tits Alternative and absence of residual finiteness in these groups.

Isometry Groups of Infinite-Genus Hyperbolic Surfaces

This paper addresses the problem of realizing specific groups as isometry groups for infinite-genus 2-manifolds without planar ends, exploring the interplay between topology and group action through hyperbolic geometry. It provides a nearly complete classification of such groups, offering insights into the constraints and possibilities of isometry groups in non-compact and non-planar settings.

Key Contributions

The central contributions can be summarized as follows:

1. Realization of Countable Groups: The authors demonstrate that for an uncountable class of infinite-genus surfaces with self-similar end spaces, every countable group can be realized as the isometry group of a complete hyperbolic metric on the surface.
2. Characterization by End Spaces: Infinite-genus surfaces are categorized based on the structure of their end spaces:
   • Self-Similar: Surfaces where the space of ends exhibits radial symmetry enable the realization of any countable group as isometries.
   • Doubly Pointed: For these surfaces, the isometry group of any complete hyperbolic metric is virtually cyclic, reflecting limited variability in geometric symmetry actions.
   • Non-Displaceable Subsurfaces: Surfaces containing such features allow only for isometry groups that are finite.
3. Algebraic Properties and Non-linear Groups: The paper explores algebraic properties of homeomorphism and mapping class groups. Notably, none of the considered groups satisfy the Tits Alternative, nor are they residually finite, cyclically orderable, or coherent. Such results underline stark contrasts with groups associated with finite-type surfaces.
4. Regular Cover Implications: The realization of groups as isometry groups has implications for regular covers of these surfaces. Particularly, surfaces whose end spaces have self-similarity support covers with deck transformation groups encompassing any countable group.

Implications and Future Directions

The classification and realization results have several implications, both theoretical and practical. Important applications extend to the paper of mapping class groups of surfaces, offering algebraic invariants that characterize surface topology. Moreover, the work furthers understanding of infinite-type surface group actions, setting grounds for extending these concepts beyond 2-dimensional surfaces.

In theoretical contexts, the paper’s methodology bridges hyperbolic geometry and topological group theory, fostering insights applicable to spaces with complex (non-compact) structures. It also prompts further inquiries into the behavior of isometries in higher-dimensional analogues and whether approaches can transcend limitations of current surface-type distinctions.

Conclusion

This paper enriches the knowledge of isometry groups for infinite-genus hyperbolic surfaces by establishing nuanced distinctions based on topology, end structure, and hyperbolic metrics. Its findings not only offer a near-complete classification but also enhance the algebraic understanding of surface groups. These results lay foundational blocks for both future theoretical exploration and practical computations in hyperbolic geometry and topology.