Deck transformations of covers of the thrice-punctured sphere #
Let c be a connected cover of the thrice-punctured sphere U = ℂ ∖ {0, 1} whose fibre over the
basepoint b = 1/2 is numbered by Fin n, and let t be its monodromy triple. A deck
transformation of c permutes the fibre over b, hence permutes the labels
(TauCeti.ConnectedFiberNumberedCover.deckPerm): the label i goes to the label of the image of
the point labelled i. This file proves that this identifies the deck group of c with the
automorphism group of t, as a subgroup of Equiv.Perm (Fin n):
deck p ≃* t.automorphismGroup.
The general input is that, over a path-connected, locally path-connected base, the permutations of
the labels induced by deck transformations are exactly the permutations commuting with the
numbered monodromy (TauCeti.ConnectedFiberNumberedCover.range_deckPerm). Over U the numbered
monodromy is generated by the components of t, and the permutations commuting with them form
t.automorphismGroup (TauCeti.PermutationTriple.automorphismGroup_eq_centralizer_monodromyGroup).
No ᵐᵒᵖ appears. Deck transformations act on the fibre on the left and commute with monodromy,
and the identification sends a deck transformation to the permutation it induces on the labels,
not to a monodromy element. The numbering is what makes the target a concrete subgroup: relabelling
the fibre by τ conjugates both sides by τ
(TauCeti.ConnectedFiberNumberedCover.deckPerm_smul,
TauCeti.PermutationTriple.automorphismGroup_smul), so only the abstract isomorphism type of the
deck group is an invariant of the unnumbered cover.
The two threaded examples are computed on the action. For the cover with triple
TauCeti.PermutationTriple.cyclicTriple n, the monodromy of z ↦ zⁿ, and for the degree-four
cover with triple TauCeti.PermutationTriple.torusTriple, the permutations of the labels induced
by deck transformations are exactly the powers of the rotation i ↦ i + 1. The deck groups are
therefore cyclic, of orders n and 4, and act transitively on the fibre; they act freely, as the
deck group of every connected cover does (TauCeti.Deck.fiber_stabilizer_eq_bot).
Main declarations #
TauCeti.ConnectedFiberNumberedCover.range_deckPerm_eq_automorphismGroup: the permutations of the labels induced by deck transformations form the automorphism group of the triple.TauCeti.ConnectedFiberNumberedCover.deckMulEquiv: the deck group of a numbered cover ofUis the automorphism group of its triple, withcoe_deckMulEquiv_applyanddeckMulEquiv_symm_smulrecording which permutation corresponds to which deck transformation.TauCeti.ConnectedFiberNumberedCover.range_deckPerm_of_connectedTriple_eq_cyclicTriple,TauCeti.ConnectedFiberNumberedCover.range_deckPerm_of_connectedTriple_eq_torusTriple: the deck actions of the two examples, with their cyclicity, their orders and their transitivity on the fibre.
References #
- E. Girondo and G. González-Diez, Introduction to Compact Riemann Surfaces and Dessins d'Enfants, London Mathematical Society Student Texts 79, Cambridge University Press, 2012, §2.7 (the monodromy and the covering group of a morphism).
- A. Hatcher, Algebraic Topology, Cambridge University Press, 2002, §1.3 (deck transformations and the lifting criterion).
The permutations of the labels induced by deck transformations of a numbered cover of
ℂ ∖ {0, 1} form the automorphism group of its triple.
The deck group of a numbered cover of ℂ ∖ {0, 1} is the automorphism group of its
triple. A deck transformation goes to the permutation it induces on the labels of the fibre over
1/2 (coe_deckMulEquiv_apply).
Equations
Instances For
The automorphism of the triple attached to a deck transformation is the permutation it induces on the labels.
The deck transformation attached to an automorphism τ of the triple sends the point labelled
i to the point labelled τ i.
The threaded examples #
Both examples induce exactly the powers of the rotation i ↦ i + 1 on the labels, and the deck
group is read off from that.
The cover of ℂ ∖ {0, 1} with the triple of z ↦ zⁿ: its deck transformations induce
exactly the powers of the rotation i ↦ i + 1 on the labels.
The deck group of the cover of ℂ ∖ {0, 1} with the triple of z ↦ zⁿ is cyclic.
The deck group of the cover of ℂ ∖ {0, 1} with the triple of z ↦ zⁿ has order n.
The deck group of the cover of ℂ ∖ {0, 1} with the triple of z ↦ zⁿ acts transitively on
the fibre over 1/2.
The degree-four cover of ℂ ∖ {0, 1} with the torus triple: its deck transformations induce
exactly the powers of the rotation i ↦ i + 1 on the labels.
The deck group of the degree-four cover of ℂ ∖ {0, 1} with the torus triple is cyclic.
The deck group of the degree-four cover of ℂ ∖ {0, 1} with the torus triple has order
4.
The deck group of the degree-four cover of ℂ ∖ {0, 1} with the torus triple acts transitively
on the fibre over 1/2; it acts freely by TauCeti.Deck.fiber_stabilizer_eq_bot.