Copyright by Aaron Joshua Fenyes 2016

245 Pages·2016·3.05 MB·English

Copyright
by
Aaron Joshua Fenyes
2016
The Dissertation Committee for Aaron Joshua Fenyes
certi(cid:28)es that this is the approved version of the following dissertation:
Warping geometric structures and
abelianizing SL R local systems
2
Committee:
Andrew Neitzke, Supervisor
David Ben-Zvi
Jacques Distler
Daniel Freed
Sean Keel
Warping geometric structures and
abelianizing SL R local systems
2
by
Aaron Joshua Fenyes, B.S.; M.S.
DISSERTATION
Presented to the Faculty of the Graduate School of
The University of Texas at Austin
in Partial Ful(cid:28)llment
of the Requirements
for the Degree of
DOCTOR OF PHILOSOPHY
THE UNIVERSITY OF TEXAS AT AUSTIN
May 2016
To: [email protected]
(To all the friends I made along the way)
Acknowledgments
This dissertation would not have been started or (cid:28)nished without the
tireless guidance of Andy Neitzke. I’m very grateful to Jen Berg for pointing
out the argument used to prove Proposition 8.2.A, and to Sona Akopian for
pointing out a key step in the proof of Theorem 9.3.K (I am, of course, re-
sponsible for any mistakes in these arguments). I’ve also enjoyed the bene(cid:28)t of
conversations, some short and some long, with Jorge Acosta, David Ben-Zvi,
Francis Bonahon, Luis Duque, Richard Hughes, Tim Magee, Tom Mainiero,
Taylor McAdam, Amir Mohammadi, Javier Morales, and Max Riestenberg, as
well as diction brainstorming with Eliana Fenyes. This research was supported
in part by NSF grants 1148490 and 1160461.
v
Warping geometric structures and
abelianizing SL R local systems
2
Publication No.
Aaron Joshua Fenyes, Ph.D.
The University of Texas at Austin, 2016
Supervisor: Andrew Neitzke
The abelianization process of Gaiotto, Hollands, Moore, and Neitzke
parameterizes SL C local systems on a punctured surface by turning them
K
intoC× localsystems, whichhaveamuchsimplermodulispace. Whenapplied
to an SL R local system describing a hyperbolic structure, abelianization pro-
2
duces an R× local system whose holonomies encode the shear parameters of
the hyperbolic structure.
This dissertation extends abelianization to SL R local systems on a
2
compact surface, using tools from dynamics to overcome the technical chal-
lenges that arise in the compact setting. Thurston’s shear parameterization
of hyperbolic structures, which has its own technical subtleties on a compact
surface, once again emerges as a special case.
vi
Table of Contents
Acknowledgments v
Abstract vi
Part I An invitation to abelianization 1
Chapter 1. Introduction 2
1.1 Context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Goals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Invitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Chapter 2. Geometric structures as local systems 4
2.1 A general framework . . . . . . . . . . . . . . . . . . . . . . . 4
2.1.1 Global to local, symmetries to sheaves . . . . . . . . . . 4
2.1.2 Geometric structures on manifolds . . . . . . . . . . . . 5
2.1.3 Analytic geometric structures . . . . . . . . . . . . . . . 8
2.1.4 Geometry unmoored . . . . . . . . . . . . . . . . . . . . 9
2.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2.1 Complex projective structures . . . . . . . . . . . . . . . 11
2.2.2 Hyperbolic structures . . . . . . . . . . . . . . . . . . . 11
2.2.3 Translation structures . . . . . . . . . . . . . . . . . . . 12
2.2.4 Half-translation structures . . . . . . . . . . . . . . . . . 13
Chapter 3. Geometric structures as (cid:29)at bundles 15
3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.2 A carrier for complex projective structures . . . . . . . . . . . 16
3.3 Carriers for hyperbolic structures . . . . . . . . . . . . . . . . 18
3.3.1 A canonical construction . . . . . . . . . . . . . . . . . 18
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3.3.2 Construction from a reference hyperbolic structure . . . 20
3.4 A general framework . . . . . . . . . . . . . . . . . . . . . . . 22
Chapter 4. Comparing geometric structures 24
4.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4.2 The Schwarzian derivative . . . . . . . . . . . . . . . . . . . . 24
4.2.1 The setup . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4.2.2 The de(cid:28)nition . . . . . . . . . . . . . . . . . . . . . . . 25
4.2.3 Showing the de(cid:28)nition makes sense . . . . . . . . . . . . 28
4.2.4 Some geometric meaning . . . . . . . . . . . . . . . . . 29
4.3 Deviations of geometric structures . . . . . . . . . . . . . . . . 33
4.4 Deviations of locally constant sheaves . . . . . . . . . . . . . . 35
Chapter 5. Warping geometric structures 37
5.1 The general idea . . . . . . . . . . . . . . . . . . . . . . . . . . 37
5.2 Cataclysms on punctured hyperbolic surfaces . . . . . . . . . . 38
5.2.1 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
5.2.2 Cataclysms on surfaces with cusps . . . . . . . . . . . . 38
5.2.3 The deviation of a cataclysm . . . . . . . . . . . . . . . 41
5.2.4 Cataclysms on surfaces with holes . . . . . . . . . . . . 43
5.3 Cataclysms on compact hyperbolic surfaces . . . . . . . . . . . 43
5.3.1 Generalizing weighted ideal triangulation . . . . . . . . 43
5.3.2 Performing cataclysms . . . . . . . . . . . . . . . . . . . 46
Chapter 6. Abelianizing geometric structures 48
6.1 Shear parameters for cusped hyperbolic surfaces . . . . . . . . 48
6.2 Shear parameters for compact hyperbolic surfaces . . . . . . . 50
6.3 Abelianization on punctured surfaces . . . . . . . . . . . . . . 51
6.3.1 A geometric realization of the shear parameters . . . . . 51
6.3.2 Shear parameters as periods . . . . . . . . . . . . . . . . 52
6.3.3 Shear parameters as holonomies . . . . . . . . . . . . . 56
6.3.4 Ideal triangulations and half-translation structures . . . 60
6.4 Abelianization on compact surfaces . . . . . . . . . . . . . . . 61
viii
Part II Abelianization on compact surfaces 63
Chapter 7. Introduction 64
7.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
7.2 Why not SL C? . . . . . . . . . . . . . . . . . . . . . . . . . . 66
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7.3 Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
7.4 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
7.4.1 Running notation . . . . . . . . . . . . . . . . . . . . . 70
7.4.2 Index of symbols . . . . . . . . . . . . . . . . . . . . . . 71
Chapter 8. Warping local systems 72
8.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
8.2 Conventions for local systems . . . . . . . . . . . . . . . . . . 72
8.2.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
8.2.2 Linear local systems . . . . . . . . . . . . . . . . . . . . 74
8.3 The descriptive power of deviations . . . . . . . . . . . . . . . 76
8.4 Warping locally constant sheaves . . . . . . . . . . . . . . . . . 78
8.5 Warping local systems . . . . . . . . . . . . . . . . . . . . . . 82
Chapter 9. Dividing translation surfaces 83
9.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
9.2 A review of translation and half-translation surfaces . . . . . . 84
9.2.1 Translation surfaces . . . . . . . . . . . . . . . . . . . . 84
9.2.2 First return maps . . . . . . . . . . . . . . . . . . . . . 87
9.2.3 Half-translation surfaces . . . . . . . . . . . . . . . . . . 90
9.3 Dividing intervals . . . . . . . . . . . . . . . . . . . . . . . . . 93
9.3.1 Construction of divided and fractured intervals . . . . . 93
9.3.2 Examples from dynamics . . . . . . . . . . . . . . . . . 94
9.3.3 Properties of divided intervals . . . . . . . . . . . . . . . 97
9.3.4 Properties of fractured intervals . . . . . . . . . . . . . . 101
9.3.5 Metrization . . . . . . . . . . . . . . . . . . . . . . . . . 103
9.4 Dividing translation surfaces . . . . . . . . . . . . . . . . . . . 108
9.4.1 Construction of divided and fractured surfaces . . . . . 108
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9.4.2 Properties of divided and fractured surfaces . . . . . . . 111
9.5 Dynamics on divided surfaces . . . . . . . . . . . . . . . . . . 114
9.5.1 The vertical (cid:29)ow . . . . . . . . . . . . . . . . . . . . . . 114
9.5.2 First return maps . . . . . . . . . . . . . . . . . . . . . 116
9.5.3 Minimality . . . . . . . . . . . . . . . . . . . . . . . . . 120
9.5.4 Ergodic theory . . . . . . . . . . . . . . . . . . . . . . . 122
9.5.5 The fat gap condition . . . . . . . . . . . . . . . . . . . 128
Chapter 10. Warping local systems on divided surfaces 135
10.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
10.2 Critical leaves in a (cid:29)ow box . . . . . . . . . . . . . . . . . . . 136
10.3 Deviations de(cid:28)ned by jumps, conceptually . . . . . . . . . . . 136
10.4 Deviations de(cid:28)ned by jumps, concretely . . . . . . . . . . . . . 137
10.4.1 The restriction property . . . . . . . . . . . . . . . . . . 138
10.4.2 The composition property . . . . . . . . . . . . . . . . . 138
Chapter 11. Uniform hyperbolicity for SL R dynamics 140
2
11.1 Motivation and notation . . . . . . . . . . . . . . . . . . . . . 140
11.2 The global version . . . . . . . . . . . . . . . . . . . . . . . . . 141
11.3 The local version . . . . . . . . . . . . . . . . . . . . . . . . . 144
11.4 The two versions are usually equivalent . . . . . . . . . . . . . 147
11.5 Extending over medians . . . . . . . . . . . . . . . . . . . . . . 151
11.6 Constructing uniformly hyperbolic local systems . . . . . . . . 153
Chapter 12. Abelianization in principle 156
12.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
12.1.1 Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
12.1.2 Framings . . . . . . . . . . . . . . . . . . . . . . . . . . 158
12.1.3 Abelianization . . . . . . . . . . . . . . . . . . . . . . . 160
12.1.4 Abelianization without punctures . . . . . . . . . . . . . 161
12.2 Running assumptions . . . . . . . . . . . . . . . . . . . . . . . 163
12.3 The slithering jump . . . . . . . . . . . . . . . . . . . . . . . . 164
12.3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . 164
x