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Some Calculations on the Action of Groups on Sur- Faces

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Some Calculations on the Action of Groups on Sur- Faces ORBIT-OnlineRepository ofBirkbeckInstitutionalTheses Enabling Open Access to Birkbeck’s Research Degree output Some calculations on the action of groups on sur- faces https://eprints.bbk.ac.uk/id/eprint/40158/ Version: Full Version Citation: Pierro, Emilio (2015) Some calculations on the action of groups on surfaces. [Thesis] (Unpublished) c 2020 The Author(s) All material available through ORBIT is protected by intellectual property law, including copy- right law. Any use made of the contents should comply with the relevant law. Deposit Guide Contact: email Some calculations on the action of groups on surfaces by Emilio Pierro Athesissubmittedto Birkbeck, University of London for the degree of Doctor of Philosophy Department of Economics, Mathematics and Statistics Birkbeck, University of London London October 17, 2015 Declaration I warrant that the content of this thesis is the direct result of my own work and that any use made in it of published or unpublished material is fully and correctly referenced. Signed ..........................................Date ............ 2 Copyright The copyright of this thesis rests with the author and no quotation from it or information derived from it may be published without the prior written consent of the author. 3 Abstract In this thesis we treat a number of topics related to generation of finite groups with motivation from their action on surfaces. The majority of our findings are presented in two chapters which can be read independently. The first deals with Beauville groups which are automorphism groups of the product of two Riemann surfaces with genus g>1, subject to some further conditions. When these two surfaces are isomorphic and transposed by elements of G we say these groups are mixed, otherwise they are unmixed. We first examine the relationship between when an almost simple group and its socle are unmixed Beauville groups and then go on to determine explicit examples of several infinite families of mixed Beauville groups. In the second we determine the 2 2m+1 2m+1 M¨obius function of the small Ree groups G2(3 )=R(3 ), where m 0, and use this to ≥ enumerate various ordered generating n-tuples of these groups. We then apply this to questions of the generation and asymptotic generation of the small Ree groups as well as interpretations in other categories, such as the number of regular coverings of a surface with a given fundamental group and whose covering group is isomorphic to R(32m+1). 4 Acknowledgment I would like to thank first and foremost my supervisor, Ben Fairbairn, for always being willing to discuss mathematics, for his time, patience and guidance throughout the course of my studies. I would like to thank (in reverse alphabetical order) Rob Wilson and Gareth Jones for taking the time to read and examine my thesis. I would also like to thank Gareth Jones for making me aware of the M¨obius function and for his encouragement throughout my PhD. I am indebted to Dimitri Leemans for our valuable conversation at SIGMAP 2014, and, after learning that I was working on the M¨obius function of the small Ree groups, for announcing this in his talk and insisting that nobody else work on this problem. I would like to thank the many mathematicians at Queen Mary, Imperial and City who have given me a mathematical community in London beyond that of my own institution. I would also like to thank my lecturers from the University of Birmingham who continue to support me long after my time there. Last but not least I wish to thank my family and friends for all of their support, in all of its forms, throughout my PhD and my whole life. I would not be here without them. The financial support I have received from Birkbeck, in particular the School Scholarship, has been life-changing and for which I cannot sufficiently express my gratitude. 5 Contents 1 Introduction 11 1.1 Beauvillegroups ...................................... 11 1.1.1 Unmixed Beauville groups . 12 1.1.2 Mixed Beauville groups . 14 1.1.3 Results ....................................... 14 1.2 The M¨obius function of a group . 16 1.2.1 Applications of the Eulerian functions of a group . 19 1.2.2 Results ....................................... 20 2 Beauville groups 21 2.1 Almost simple unmixed Beauville groups . 22 2.1.1 Outer automorphism groups that are not 2-generated . 23 2.1.2 Non-split extensions . 24 2.1.3 Splitextensions................................... 26 2.1.4 Examples ...................................... 27 2.2 Topological invariants of Beauville surfaces . 28 2.3 Mixed Beauville groups via mixable Beauville structures . 30 2.4 Mixable Beauville structures for the alternating groups . 33 2.5 Mixable Beauville structures for low-rank finite groups of Lie type . 37 2.5.1 The projective special linear groups L2(q).................... 38 2.5.2 The projective special linear groups L3(q).................... 41 2.5.3 The projective special unitary groups U3(q)................... 44 2.5.4 The projective symplectic groups S4(q)..................... 47 2.5.5 The Suzuki groups Sz(22m+1)........................... 51 2.5.6 The exceptional groups G2(q)........................... 51 2.5.7 The small Ree groups R(32m+1)......................... 54 2 2m+1 2.5.8 The large Ree groups F4(2 )......................... 55 3 2.5.9 The Steinberg triality groups D4(q)....................... 56 2.6 Mixable Beauville structures for the sporadic groups . 57 6 3 The M¨obius function of the small Ree groups 61 3.1 The M¨obius function of R(3) ............................... 63 3.2 ThestructureofthesimplesmallReegroups . 65 3.2.1 Conjugacy classes and centralisers of elements in R(q)............. 65 3.2.2 Maximal subgroups . 67 3.3 The M¨obius function of the simple small Ree groups . 70 3.3.1 Conjugacy classes and normalisers of subgroups in R(q)............ 71 3.3.2 Intersections of maximal subgroups . 73 3.4 Eulerian functions of the small Ree groups . 85 3.4.1 Enumerations of Epi(Γ,R(3))........................... 87 3.4.2 Free groups & Hecke groups . 88 3.4.3 Asymptoticresults ................................. 91 3.5 General results on the M¨obius and Eulerian functions . 92 A The genus spectrum of a group 94 A.1 Program........................................... 94 A.2 The genus spectrum of some finite almost simple groups . 97 B Future research 111 B.1 Beauvillegroups ...................................... 111 B.1.1 Mixable Beauville groups . 111 B.1.2 Beauville p-groups ................................. 112 B.1.3 Strongly real Beauville groups . 112 B.2 M¨obiusfunctionsandrelatedproblems. 113 B.2.1 MoreM¨obiusfunctions. .. 113 B.2.2 Other properties of subgroup lattices . 114 B.2.3 Maximal subgroups . 115 7 List of Tables 2.1 Words in the standard generators for the automorphism groups of the sporadic groups andtheTitsgroup...................................... 28 2.2 Power maps for elements of generating triples in Table 2.1. 29 2.3 Mixable Beauville structures for G and G G where G = U (q), q =4, 5 and 8. 47 ⇥ 3 2.4 Mixable strucutres (x1,y1; x2,y2) for G in terms of words in the standard generators. 59 2.5 Types of the mixable Beauville structures for G and G G from the words in Table ⇥ 2.4 and Lemmas 2.87 and 2.88. 60 3.1 Table of values for ⌫K (H) with the isomorphism class of H in the left hand column. 65 3.2 Conjugacy classes of maximal subgroups of the simple small Ree groups R(q). 67 3.3 The disjoint subsets of MaxInt. Each subset consists of subgroups of G for all l dividing n unlessotherwisestated. 70 3.4 H = R(3h) R(l)..................................... 75 ⇠ 2 3.5 H = (3h)1+1+1 :(3h 1) P(l).............................. 75 ⇠ − 2 3.6 H = 2 L (3h) C (l).................................. 76 ⇠ ⇥ 2 2 t h 3h 1 ! 3.7 H = 2 (3 : − ) C (l)................................ 77 ⇠ ⇥ 2 2 t 3.8 H = 3h 1 C (l).................................... 79 ⇠ − 2 0 2 3.9 H = (2 D h ):3 N (l)............................. 82 ⇠ ⇥ (3 +1)/2 2 V h h+1 3.10 H = 3 3 2 +1:6 N (l)............................... 82 ⇠ − 2 2 2 3.11 H = 2 D h C (l)............................... 83 ⇠ ⇥ (3 +1)/2 2 V 3.12 H = D D (l).................................... 83 ⇠ 2a2(h) 2 2 3.13 H = 2 L (3) C (1)................................... 83 ⇠ ⇥ 2 2 t 3.14 H = 23 E ......................................... 84 ⇠ 2 3.15 H = 22 V ......................................... 84 ⇠ 2 3.16 H = tu C ⇤ ....................................... 85 ⇠ h i2 6 3.17 H = u C ⇤ ........................................ 85 ⇠ h i2 3 3.18 H = t C ........................................ 86 ⇠ h i2 2 3.19 H I ............................................ 86 2 3.20 Values of H for nontrivial subgroups of R(3) with µ (H) = 0. 88 | |n G 6 8 3.21 Evaluation of da,b(R(3))................................... 88 3.22 Values of H for n =2, 3or6............................... 89 | |n 3.23 Values of H for n =7or9................................ 89 | |n 3.24 Values of d (G) for R(q), q 39.............................. 90 2 3.25 Values of h (G) for R(q), q 39.............................. 91 3 A.1 Genus spectrum of A6 ................................... 98 A.2 Genus spectrum of S6 ................................... 98 A.3 Genus spectrum of A7 ................................... 99 A.4 Genus spectrum of S7, continued on Table A.5 . 100 A.5 Genus spectrum of S7 continued ............................. 101 A.6 Genus spectrum of P ΓL2(8) ⇠= R(3) ........................... 102 A.7 Genus spectrum of L2(16): 2 ⇠= O4−(4) .......................... 102 A.8 Genus spectrum of P ΓL2(16) ............................... 102 A.9 Genus spectrum of P ⌃L2(25), continued on Table A.10 . 103 A.10 Genus spectrum of P ⌃L2(25) continued . 104 A.11 Genus spectrum of P ⌃L2(27)..............................
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