Alternating Quotients of Fuchsian Groups

Everitt, Brent

doi:10.1006/jabr.1999.8014

Cited by 43 publications

(42 citation statements)

References 14 publications

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“…Everitt (2000), [Ev00]) asserting that every hyperbolic triangle group surjects onto almost all alternating groups. The next result gives explicit examples of rigid surfaces not biholomorphic to their complex conjugate surface.…”

Section: In Particular S(v) Is Isomorphic To S(v) If and Only If S(v)mentioning

confidence: 99%

Chebycheff and Belyi Polynomials, Dessins d’Enfants, Beauville Surfaces and Group Theory

Bauer

Catanese

Grunewald

2006

MedJM

161

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Abstract. We start discussing the group of automorphisms of the field of complex numbers, and describe, in the special case of polynomials with only two critical values, Grothendieck's program of 'Dessins d' enfants', aiming at giving representations of the absolute Galois group. We describe Chebycheff and Belyi polynomials, and other explicit examples. As an illustration, we briefly treat difference and Schur polynomials. Then we concentrate on a higher dimensional analogue of the triangle curves, namely, Beauville surfaces and varieties isogenous to a product. We describe their moduli spaces, and show how the study of these varieties leads to new interesting questions in the theory of finite (simple) groups.

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Section: In Particular S(v) Is Isomorphic To S(v) If and Only If S(v)mentioning

confidence: 99%

Chebycheff and Belyi Polynomials, Dessins d’Enfants, Beauville Surfaces and Group Theory

Bauer

Catanese

Grunewald

2006

MedJM

161

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“…It is hoped to give full details of this proof in a later paper, together with some applications of maximal subgroups. It seems plausible that coset diagrams constructed by Everitt [4] and others could be used to extend this result to other triangle groups, and to more general Fuchsian groups. In particular, Theorem 1 and Proposition 11 suggest: Conjecture 12.…”

Section: Consequences and Generalisationsmentioning

confidence: 79%

Maximal subgroups of the modular and other groups

Jones

2018

Journal of Group Theory

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In 1933 B. H. Neumann constructed uncountably many subgroups of SL 2 (Z) which act regularly on the primitive elements of Z 2 . As pointed out by Magnus, their images in the modular group PSL 2 (Z) C 3 * C 2 are maximal nonparabolic subgroups, that is, maximal with respect to containing no parabolic elements. We strengthen and extend this result by giving a simple construction using planar maps to show that for all integers p ≥ 3, q ≥ 2 the triangle group Γ = ∆(p, q, ∞) C p * C q has uncountably many conjugacy classes of nonparabolic maximal subgroups. We also extend results of Tretkoff and of Brenner and Lyndon for the modular group by constructing uncountably many conjugacy classes of such subgroups of Γ which do not arise from Neumann's original method. These maximal subgroups are all generated by elliptic elements, of finite order, but a similar construction yields uncountably many conjugacy classes of torsion-free maximal subgroups of the Hecke groups C p * C 2 for odd p ≥ 3. Finally, an adaptation of work of Conder yields uncountably many conjugacy classes of maximal subgroups of ∆(2, 3, r) for all r ≥ 7.

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“…This establishes a conjecture of Higman (in the oriented case), which generalises Conder's theorem on Hurwitz groups (see Theorem 2.10). Everitt's approach in [25] builds on the coset-diagram methodology developed by Higman and Conder. By applying very different methods, using a combination of character-theoretic and probabilistic tools, Liebeck and Shalev prove the following theorem, which settles Higman's conjecture in full generality (see [47,Theorem 1.7]).…”

Section: Triangle Generationmentioning

confidence: 99%

Simple Groups, Generation and Probabilistic Methods

Burness¹

2019

Groups St Andrews 2017 in Birmingham

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It is well known that every finite simple group can be generated by two elements and this leads to a wide range of problems that have been the focus of intensive research in recent years. In this survey article we discuss some of the extraordinary generation properties of simple groups, focussing on topics such as random generation, (a, b)-generation and spread, as well as highlighting the application of probabilistic methods in the proofs of many of the main results. We also present some recent work on the minimal generation of maximal and second maximal subgroups of simple groups, which has applications to the study of subgroup growth and the generation of primitive permutation groups.

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Alternating Quotients of Fuchsian Groups

Abstract: It is shown that any finitely generated, non-elementary Fuchsian group has among its homomorphic images all but finitely many of the alternating groups A n . This settles in the affirmative a long-standing conjecture of Graham Higman.

Cited by 43 publications

References 14 publications

Chebycheff and Belyi Polynomials, Dessins d’Enfants, Beauville Surfaces and Group Theory

Chebycheff and Belyi Polynomials, Dessins d’Enfants, Beauville Surfaces and Group Theory

Maximal subgroups of the modular and other groups

Simple Groups, Generation and Probabilistic Methods

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