Géométrie et Topologie, Théorie des groupes

Free actions on surfaces that do not extend to arbitrary actions on 3-manifolds

Samperton, Eric G.¹  

¹ Department of Mathematics, University of Illinois, 1409 West Green Street (MC-382), Urbana, Illinois 61801, USA

Comptes Rendus. Mathématique, Tome 360 (2022) no. G2, pp. 161-167.

Résumé

We provide the first known example of a finite group action on an oriented surface $T$ that is free, orientation-preserving, and does not extend to an arbitrary (in particular, possibly non-free) orientation-preserving action on any compact oriented 3-manifold $N$ with boundary $\partial N=T$. This implies a negative solution to a conjecture of Domínguez and Segovia, as well as Uribe’s evenness conjecture for equivariant unitary bordism groups. We more generally provide sufficient conditions implying that infinitely many such group actions on surfaces exist. Intriguingly, any group with such a non-extending action is also a counterexample to the Noether problem over the complex numbers $ℂ$. In forthcoming work with Segovia we give a complete homological characterization of those finite groups admitting such a non-extending action, as well as more examples and non-examples. We do not address here the analogous question for non-orientation-preserving actions.

Détail

Reçu le : 2021-07-02
Révisé le : 2021-07-30
Accepté le : 2021-09-29
Publié le : 2022-02-15

DOI : 10.5802/crmath.277

Classification : 57M60, 57M10

Licence : CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

Bibliographie

[1] Bogomolov, Fedor A. The Brauer group of quotient spaces of linear representations, Izv. Akad. Nauk SSSR, Ser. Mat., Volume 51 (1987) no. 3, 688, pp. 485-516 | DOI

[2] Brand, Neal Classifying spaces for branched coverings, Indiana Univ. Math. J., Volume 29 (1980) no. 2, pp. 229-248 | MR | Zbl | DOI

[3] Cooper, Daryl; Hodgson, Craig D.; Kerckhoff, Steven P. Three-dimensional orbifolds and cone-manifolds, MSJ Memoirs, 5, Mathematical Society of Japan, 2000 (with a postface by Sadayoshi Kojima) | Zbl

[4] Domínguez, Jesús Emilio; Segovia, Carlos Extending free actions of finite groups on surfaces, Topology Appl., Volume 305 (2022), 107898 | MR | Zbl | DOI

[5] Ellenberg, Jordan S.; Venkatesh, Akshay; Westerland, Craig Homological stability for Hurwitz spaces and the Cohen–Lenstra conjecture over function fields, II (2013) (https://arxiv.org/abs/1212.0923v2)

[6] Gradolato, Monique; Zimmermann, Bruno Extending finite group actions on surfaces to hyperbolic $3$-manifolds, Math. Proc. Camb. Philos. Soc., Volume 117 (1995) no. 1, pp. 137-151 | MR | Zbl | DOI

[7] Hidalgo, Rubén A. On Schottky groups with automorphism, Ann. Acad. Sci. Fenn., Math., Volume 19 (1994) no. 2, pp. 259-289 | MR | Zbl

[8] Hoshi, Akinari; Kang, Ming-Chang; Kunyavskii, Boris E. Noether’s problem and unramified Brauer groups, Asian J. Math., Volume 17 (2013) no. 4, pp. 689-713 | MR | Zbl | DOI

[9] Jezernik, Urban; Moravec, Primož Bogomolov multipliers of groups of order 128, Exp. Math., Volume 23 (2014) no. 2, pp. 174-180 | MR | Zbl | DOI

[10] Kunyavskiĭ, Boris The Bogomolov multiplier of finite simple groups, Cohomological and geometric approaches to rationality problems. New Perspectives (Bogomolov, Fedor et al., eds.) (Progress in Mathematics), Volume 282, Birkhäuser, 2010, pp. 209-217 | MR | Zbl | DOI

[11] Moravec, Primož Unramified Brauer groups of finite and infinite groups, Am. J. Math., Volume 134 (2012) no. 6, pp. 1679-1704 | MR | Zbl | DOI

[12] Ossa, Erich Unitary bordism of abelian groups, Proc. Am. Math. Soc., Volume 33 (1972), pp. 568-571 | MR | Zbl | DOI

[13] Reni, Marco; Zimmermann, Bruno Extending finite group actions from surfaces to handlebodies, Proc. Am. Math. Soc., Volume 124 (1996) no. 9, pp. 2877-2887 | MR | Zbl | DOI

[14] Rowlett, Russell J. Bordism of metacyclic group actions, Mich. Math. J., Volume 27 (1980) no. 2, pp. 223-233 | MR | Zbl

[15] Saltman, David J. Noether’s problem over an algebraically closed field, Invent. Math., Volume 77 (1984) no. 1, pp. 71-84 | MR | Zbl | DOI

[16] Samperton, Eric Schur-type invariants of branched $G$-covers of surfaces, Topological phases of matter and quantum computation. AMS special session, Bowdoin College, Brunswick, ME, USA, September 24–25, 2016 (Bruillard, Paul et al., eds.) (Contemporary Mathematics), Volume 747, American Mathematical Society, 2020, pp. 173-197 | MR | Zbl | DOI

[17] Stong, Robert E. Unoriented bordism and actions of finite groups, Memoirs of the American Mathematical Society, 103, American Mathematical Society, 1970 | Zbl

[18] The GAP Group GAP – Groups, Algorithms, and Programming, 2021 (Version 4.11.1, https://www.gap-system.org)

[19] Uribe, Bernardo The evenness conjecture in equivariant unitary bordism, Proceedings of the International Congress of Mathematicians—Rio de Janeiro 2018. Vol. II. Invited lectures (Sirakov, Boyan et al., eds.), World Scientific; Sociedade Brasileira de Matemática, 2018, pp. 1217-1239 | Zbl

[20] Uribe, Bernardo Private communication, 2021