Quotient triples by blocks #
Let t be a permutation triple and B a set of sheets. The monodromy group of t permutes the
translates g • B of B, so after numbering those translates by Fin m the three components of
t induce a triple of degree m: this is TauCeti.PermutationTriple.blockQuotient. It is always
connected, since a group acts transitively on each of its orbits, and changing the numbering only
relabels it.
When t is connected and B is a nonempty block of the monodromy action, the translates of B
partition the sheets (MulAction.IsBlock.isBlockSystem), and sending a sheet to the number of the
translate containing it is a surjection TauCeti.PermutationTriple.blockIndex from the sheets of
t to those of the quotient which intertwines the three components. Geometrically, the cover
described by t factors through the cover described by the quotient triple, and the fibres of
the intermediate map are the blocks. The degree of the quotient times the size of B is the
degree of t.
The cycle data of t refines that of the quotient. A cycle of an element g of the monodromy
group goes round the cycle of g on the blocks below it a whole number of times, namely the
number of its sheets lying in one block. Counting cycles, g has at most |B| times as many
cycles on the sheets as on the blocks; summed over the three components this bounds the Euler
characteristic of t by |B| times that of the quotient, which is the combinatorial form of the
Riemann–Hurwitz inequality for the intermediate cover, and shows that passing to a quotient does
not increase the genus.
Every block system of a transitive action that is stable under the group consists of the
translates of any one of its blocks, so describing quotients through a single block B loses
nothing.
Quotients are transitive. The blocks of the quotient by B containing the translate B itself
correspond, by pulling back along the quotient map, to the blocks of t containing B, so the
block systems of the quotient are exactly the block systems of t coarser than the translates of
B. Taking the quotient of the quotient by a set C of its sheets is taking the quotient of t
by the preimage of C, and the quotient maps compose accordingly: geometrically, a tower of
intermediate covers is read off from a chain of block systems.
Main definitions #
TauCeti.PermutationTriple.blockActionHom: the action of the monodromy group on the translates ofB, numbered byFin m.TauCeti.PermutationTriple.blockQuotient: the quotient triple.TauCeti.PermutationTriple.blockIndex: the quotient map on sheets.TauCeti.PermutationTriple.preimageBlockIndexOrderIso: the blocks of the quotient containingBare the blocks oftcontainingB.TauCeti.PermutationTriple.blockIndexOrbitEquiv: the translates of a set of sheets of the quotient are the translates of its preimage.
Main results #
TauCeti.PermutationTriple.blockQuotient_eq_smul: two numberings of the translates give quotient triples related by an explicit relabeling.TauCeti.PermutationTriple.isConnected_blockQuotient: quotient triples are connected.TauCeti.PermutationTriple.blockIndex_σ0,blockIndex_σ1,blockIndex_σinf: the quotient map intertwines the components oftwith those of the quotient.TauCeti.PermutationTriple.ncard_mul_eq_of_isBlock: the size of the block times the degree of the quotient is the degree oft.TauCeti.PermutationTriple.isBlock_preimage_blockIndex_iff: a set of sheets of the quotient is a block exactly when its preimage is one.TauCeti.PermutationTriple.blockQuotient_blockQuotient: the quotient of the quotient byCis the quotient by the preimage ofC.TauCeti.PermutationTriple.equivalent_blockQuotient_image_blockIndex: the quotient by a block containingBis, up to isomorphism, a quotient of the quotient byB.TauCeti.PermutationTriple.blockIndex_blockIndex: the quotient maps compose.TauCeti.PermutationTriple.ncard_sameCycle_and_mem_mul_minimalPeriod_blockActionHom: the length of the cycle of a sheet is the length of the cycle of its block times the number of sheets of that cycle in the block.TauCeti.PermutationTriple.eulerChar_le_ncard_mul_eulerChar_blockQuotient: the Euler characteristic oftis at most|B|times that of the quotient.TauCeti.PermutationTriple.genus_blockQuotient_le: the quotient of a connected triple has genus at most that of the triple.
References #
- S. K. Lando, A. K. Zvonkin, Graphs on Surfaces and Their Applications, Encyclopaedia of Mathematical Sciences 141, Springer 2004, §1.5.
- J. D. Dixon and B. Mortimer, Permutation Groups, Graduate Texts in Mathematics 163, Springer 1996, §1.5.
The quotient triple #
The action of the monodromy group of t on the translates of B, with the translates
numbered by e.
Equations
- t.blockActionHom B e = e.permCongrHom.toMonoidHom.comp (MulAction.toPermHom ↥t.monodromyGroup ↑(MulAction.orbit (↥t.monodromyGroup) B))
Instances For
Numbering the translates of B by e, the permutation of the numbers induced by g is
the action of g on the translates.
The quotient of a triple by the translates of B: the triple of permutations induced on
the translates, numbered by e, by the three components.
Equations
- t.blockQuotient B e = t.mapMonodromy (t.blockActionHom B e)
Instances For
The first component of the quotient moves the translate numbered i by t.σ0.
The second component of the quotient moves the translate numbered i by t.σ1.
The third component of the quotient moves the translate numbered i by t.σinf.
The monodromy group of the quotient is the image of the monodromy group of t acting on
the translates of B.
Changing the numbering of the translates from e to e' relabels the quotient triple by
the permutation e.symm.trans e' of Fin m comparing the two numberings.
The quotient triple does not depend on the numbering of the translates, up to isomorphism.
A quotient triple is connected: it has a sheet, the translate B itself, and the monodromy
group of t acts transitively on the translates of B.
The quotient map on sheets #
For a triple with transitive monodromy and a nonempty block B, the number of the unique
translate of B containing the sheet x.
Equations
- TauCeti.PermutationTriple.blockIndex ht hB hBne e x = e ((hB.imprimitivityEquiv hBne) x).1
Instances For
The sheet x is sent to i exactly when it lies in the translate numbered i.
Every sheet lies in the translate of B whose number it is sent to.
Every sheet of the quotient is hit: translates of a nonempty set are nonempty.
The quotient map is equivariant for the monodromy group, acting on the quotient through
TauCeti.PermutationTriple.blockActionHom.
The quotient map intertwines the first components.
The quotient map intertwines the second components.
The quotient map intertwines the third components.
The degree of a triple with transitive monodromy is the size of a nonempty block times the degree of the quotient by it.
Composing quotients #
The action of an element of the monodromy group of t on the translates of B lies in the
monodromy group of the quotient.
Pulling back along the quotient map turns the translate of a set of sheets of the quotient
by h into the translate of its preimage by any g acting on the translates of B as h.
Blocks of a quotient. A set of sheets of the quotient is a block of its monodromy action
exactly when its preimage under the quotient map is a block of the monodromy action of t.
Blocks containing B come from the quotient. A block of the monodromy action of t
containing B is the preimage of its image under the quotient map: it is a union of translates
of B.
Block systems refine. Pulling back along the quotient map is an order isomorphism from
the blocks of the monodromy action of the quotient containing the sheet that numbers the
translate B itself to the blocks of the monodromy action of t containing B.
Equations
- One or more equations did not get rendered due to their size.
Instances For
The block of t corresponding to a block of the quotient is its preimage.
The block of the quotient corresponding to a block of t containing B is its image.
Pulling back along the quotient map identifies the translates of a set C of sheets of the
quotient with the translates of its preimage.
Equations
- One or more equations did not get rendered due to their size.
Instances For
The translate of the preimage of C corresponding to a translate of C is its preimage.
The translate of C corresponding to a translate of its preimage is its image.
The identification of translates is equivariant for the monodromy group of t, acting on
the translates of C through its action on the sheets of the quotient.
The action on the translates of C of the quotient, through the action on the sheets of the
quotient, is the action on the translates of the preimage of C, numbered compatibly.
Quotient triples compose. Taking the quotient of the quotient of t by B by a set C
of its sheets gives the quotient of t by the preimage of C, for the numbering of the
translates of the preimage induced by that of the translates of C.
Up to isomorphism, the quotient of the quotient of t by B by a set C of its sheets is
the quotient of t by the preimage of C, whatever the numberings of the translates.
Quotients by coarser blocks factor through finer ones. For a block D of the monodromy
action of t containing B, the quotient of t by D is, up to isomorphism, the quotient of
the quotient of t by B by the image of D.
Quotient maps compose. For a nonempty block C of the monodromy action of the quotient
of t by B, the quotient map of t by the preimage of C is the quotient map of t by B
followed by the quotient map of the quotient by C.
Cycle data of the quotient #
The quotient map is a semiconjugacy from each element of the monodromy group to its action on the translates.
Each fibre of the quotient map is a translate of B, so it has as many sheets as B.
Cycle lengths in a block quotient. For g in the monodromy group and a sheet x, the
length of the cycle of x under g is the length of the cycle of the block of x under the
quotient action of g, times the number of sheets of the cycle of x that lie in the block
of x.
An element of the monodromy group has at most |B| times as many cycles on the sheets as on
the translates of B.
The Riemann–Hurwitz inequality for a block quotient. The Euler characteristic of a triple
with transitive monodromy is at most |B| times that of its quotient by a nonempty block B.
Passing to a block quotient does not increase the genus. The quotient of a connected triple by a nonempty block has genus at most that of the triple.