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Canonical decompositions of hyperbolic 3-orbifolds

Fincher, Mark (2022) Canonical decompositions of hyperbolic 3-orbifolds. Doctoral Dissertation, University of Pittsburgh. (Unpublished)

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This thesis describes the theory behind Sym, software created by the author for computations
with finite-volume cusped hyperbolic 3-orbifolds. The main purpose of Sym, in
its current form, is to compute canonical (Epstein-Penner) decompositions of these orbifolds.
This was originally motivated by a joint project between the author, his advisor, and
his advisor’s other graduate students to create a census of orbifolds commensurable to the
figure-eight knot complement.
Underlying Sym is a non-standard notion of an orbifold triangulation, in which tetrahedra
may be labeled with groups of symmetries acting on them. This allows us to consider
fully ideal hyperbolic triangulations of orbifolds, which we attempt to treat in the same
way that SnapPy treats ideal triangulations of manifolds. SnapPy is powerful existing software
for hyperbolic 3-manifolds and some orbifolds, originally developed by Weeks and now
maintained by Culler, Dunfield, and Goerner.
The way SnapPy finds canonical decompositions of hyperbolic manifolds is complicated
both theoretically and computationally, and relies on influential work by Epstein, Penner,
Weeks, and others. The main goal of this thesis is to extend that work to orbifolds. A key
idea we develop is an orbifold version of Pachner moves, which are moves which change an
orbifold triangulation locally.


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Item Type: University of Pittsburgh ETD
Status: Unpublished
CreatorsEmailPitt UsernameORCID
Fincher, Markmef98@pitt.edumef98
ETD Committee:
TitleMemberEmail AddressPitt UsernameORCID
Committee ChairDeBlois,
Committee MemberDunfield,
Committee MemberWang-Erickson,
Committee MemberSchikorra,
Date: 30 September 2022
Date Type: Publication
Defense Date: 2 August 2022
Approval Date: 30 September 2022
Submission Date: 4 August 2022
Access Restriction: No restriction; Release the ETD for access worldwide immediately.
Number of Pages: 95
Institution: University of Pittsburgh
Schools and Programs: Dietrich School of Arts and Sciences > Mathematics
Degree: PhD - Doctor of Philosophy
Thesis Type: Doctoral Dissertation
Refereed: Yes
Uncontrolled Keywords: Orbifold, Hyperbolic geometry, Canonical decomposition
Date Deposited: 30 Sep 2022 18:57
Last Modified: 30 Sep 2022 18:57

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