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Arbitrarily large Galois orbits of non-homeomorphic surfaces

Arbitrarily large Galois orbits of non-homeomorphic surfaces
Arbitrarily large Galois orbits of non-homeomorphic surfaces

It has recently been proved that any element σ of the absolute Galois group GalQ¯/Q different from the identity and complex conjugation conjugates some surface S into a surface Sσ with non-isomorphic fundamental group. Here we use group theory to construct, for each integer n⩾ 0 , an n-dimensional family of complex surfaces whose conjugates under GalQ¯/Q exhibit arbitrarily many non-isomorphic fundamental groups. These fundamental groups nevertheless have isomorphic profinite completions. The surfaces constructed are isogenous to higher products via actions of groups PGL 2(p) on algebraic curves.

Algebraic and topological fundamental groups, Beauville surfaces, Galois orbits, Triangle groups
2199-675X
223-241
González-Diez, Gabino
f8de8eec-3c08-48e8-aeaf-6e77e59dcc1d
Jones, Gareth A.
fdb7f584-21c5-4fe4-9e57-b58c78ebe3f5
Torres-Teigell, David
5d1152bb-9731-4ad0-aed8-79b25050a344
González-Diez, Gabino
f8de8eec-3c08-48e8-aeaf-6e77e59dcc1d
Jones, Gareth A.
fdb7f584-21c5-4fe4-9e57-b58c78ebe3f5
Torres-Teigell, David
5d1152bb-9731-4ad0-aed8-79b25050a344

González-Diez, Gabino, Jones, Gareth A. and Torres-Teigell, David (2018) Arbitrarily large Galois orbits of non-homeomorphic surfaces. European Journal of Applied Mathematics, 4 (1), 223-241. (doi:10.1007/s40879-017-0203-z).

Record type: Article

Abstract

It has recently been proved that any element σ of the absolute Galois group GalQ¯/Q different from the identity and complex conjugation conjugates some surface S into a surface Sσ with non-isomorphic fundamental group. Here we use group theory to construct, for each integer n⩾ 0 , an n-dimensional family of complex surfaces whose conjugates under GalQ¯/Q exhibit arbitrarily many non-isomorphic fundamental groups. These fundamental groups nevertheless have isomorphic profinite completions. The surfaces constructed are isogenous to higher products via actions of groups PGL 2(p) on algebraic curves.

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Accepted/In Press date: 15 November 2017
e-pub ahead of print date: 29 November 2017
Published date: 1 March 2018
Keywords: Algebraic and topological fundamental groups, Beauville surfaces, Galois orbits, Triangle groups

Identifiers

Local EPrints ID: 418885
URI: https://eprints.soton.ac.uk/id/eprint/418885
ISSN: 2199-675X
PURE UUID: 6035cf27-3e78-45b5-95d1-c0d98d4818f2

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Date deposited: 23 Mar 2018 17:30
Last modified: 14 Mar 2019 05:11

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