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Bounds on the orders of groups of automorphisms of a pseudo‐real surface of given genus
Author(s) -
Bujalance E.,
Cirre F. J.,
Conder M. D. E.
Publication year - 2020
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms.12296
Subject(s) - automorphism , genus , mathematics , surface (topology) , pure mathematics , group (periodic table) , combinatorics , geometry , physics , zoology , biology , quantum mechanics
A compact Riemann surface is called pseudo‐real if it admits anti‐conformal (orientation‐reversing) automorphisms, but no anti‐conformal automorphism of order 2 . In this paper, we consider upper bounds on the order of a group G of automorphisms of a pseudo‐real surface S of given genus g > 1 , in general and for certain special cases. We determine for all g ≥ 2 the orders of the largest cyclic group and the largest abelian group of automorphisms of a pseudo‐real surface of genus g , containing orientation‐reversing elements, and consider the problem of finding similar bounds when the group contains no orientation‐reversing elements. For arbitrary groups, we show that if M ( g ) is the order of the largest group of automorphisms of a pseudo‐real surface of genus g , then M ( g ) ≥ 2 g for every even g ≥ 2 , while M ( g ) ≥ 4 ( g − 1 ) for every odd g ≥ 3 , and we prove that the latter bound is sharp for a very large and possibly infinite set of odd values of g ≥ 3 . We also give the precise values of M ( g ) for all g between 2 and 128 , together with the signatures for the actions of the corresponding groups of largest order.