Polynomial automorphisms of C^n preserving the Markoff-Hurwitz polynomial

0.66 MB·English

by  Hengnan Hu

POLYNOMIAL AUTOMORPHISMS OF Cn PRESERVING THE
MARKOFF-HURWITZ POLYNOMIAL
5 HENGNANHU,SERPEOWTAN,ANDYINGZHANG
1
0 To Professor William Goldman on the occasion of his sixtieth birthday
2

y Abstract. We study the action of the group of polynomial automorphisms
a ofCn (n≥3)whichpreservetheMarkoff-Hurwitzpolynomial
M
H(x):=x21+x22+···+x2n−x1x2···xn.
Ourmainresultsincludethedeterminationofthegroup,thedescriptionofa
6
non-emptyopensubsetofCnonwhichthegroupactsproperlydiscontinuously

(domainofdiscontinuity),andidentitiesfortheorbitofpointsinthedomain
]
T ofdiscontinuity.
G
.
h
t
a Contents
m
1. Introduction 2
[
2. Combinatorial setting and basic results 8

2 2.1. Cayley graphs and orbits 8
v 2.2. Index sets [n], P(n), Iˆ, {ˆi} and C(n,k) 8
5
5 2.3. Rooted regular trees 9
9 2.4. Regular subtrees: T|k|(∆), X :=T|n−1|(∆) and A:=T|2|(∆) 9
6 2.5. Hurwitz maps 10
0
2.6. Dihedral Hurwitz maps 11
.
1 2.7. Extended Hurwitz maps 11
0 2.8. Directed trees 11
5
2.9. Sinks, merges and forks 12
1
2.10. Induced directed tree ∆(cid:126) 12
: φ
v 2.11. Edge relations for alternating geodesics 12
i
X 2.12. Edge-connectedness of A (K) 13
φ
2.13. Circular sets 14
r
a 2.14. Subsets A0(e), A−((cid:126)e) and A0−((cid:126)e) of A 14
2.15. The Fibonacci function F on A 15
2.16. Sierpinski simplices 16
2.17. Growth along an alternating geodesic 17
2.18. Sink estimates 18
3. Bowditch conditions and domain of discontinuity 19
2000 Mathematics Subject Classification. 57M05;32G15;30F60;20H10;37F30.
Key words and phrases. Markoff-Hurwitz polynomial, Cayley graph, polynomial automor-
phisms,identities.
HuandTanarepartiallysupportedbytheNationalUniversityofSingaporeacademicresearch
grant R-146-000-186-112. Zhang is supported by NSFC (China) grant no. 11271276 and Ph.D.
ProgramsFoundation(China)grantno. 20133201110001.
1
2 H.HU,S.P.TAN,ANDY.ZHANG
3.1. The Bowditch conditions 19
3.2. Escaping ray 19
3.3. Attracting subtrees 21
3.4. Attracting subtree T 21
φ
3.5. Fibonacci growth 24
3.6. Enlarged attracting subtree 26
3.7. Openness of B 26
3.8. The diagonal slice 27
3.9. The complement of D in Cn 28
3.10. Properly discontinuous action of Γ on D 28
n
4. Identities and proofs 28
4.1. Convergence of infinite sums 29
4.2. Induced weight ψ((cid:126)e) on a directed edge 30
4.3. Solving for ψ((cid:126)e) 30
4.4. Estimate of ψ((cid:126)e)− 1h(φ(γ)) 31
2
4.5. Proofs of identities 31
4.6. Variations of identities 33
5. Further directions 34
Appendix: Polynomial automorphisms 34
References 36
1. Introduction
The Markoff equation
x2+x2+x2−x x x =0 (1)
1 2 3 1 2 3
occurs in various settings and has been well studied by many authors in different
contexts (actually Markoff studied the equation a2 +b2 +c2 = 3abc, the triples
of positive integer solutions are called Markoff triples, but this is equivalent to
(1) by the change of variables x = 3a,x = 3b,x = 3c). As a diophantine
1 2 3
equation, the triples of positive integer solutions have interpretations in terms of
diophantineapproximations,minimaofbinaryquadraticformsandtracesofsimple
closed geodesics on the modular torus, see [18] for an excellent survey. A basic
theorem of Markoff states that the set of (non-trivial) integer solutions of (1) can
beobtainedfromthefundamentalsolution(3,3,3)byconsideringitsorbitunderthe
actionofthegroupΓ∗ofpolynomialautomorphismsof(1). HereΓ∗isgeneratedby
3 3
thepermutations, the evensign changeautomorphisms (wheretwoof thevariables
change signs) and the involution b :C3 →C3 given by
1
b (x ,x ,x )=(x x −x ,x ,x ).
1 1 2 3 2 3 1 2 3
More generally, C3 parametrizes
Hom(F ,SL(2,C))//SL(2,C),
2
thecharactervarietyofthefreegroupontwogeneratorsF =(cid:104)A,B(cid:105)intoSL(2,C),
2
where the parameters are x = trA, x = trB and x = tr(AB). The variety
1 2 3
AUTOMORPHISMS PRESERVING THE MARKOFF-HURWITZ POLYNOMIAL 3
described by (1) parametrizes the relative character variety
{[ρ]∈Hom(F ,SL(2,C))//SL(2,C) | tr(ABA−1B−1)=−2},
2
see [3, 6, 20, 21]. In this context, Γ∗ is commensurable to the action of Out(F ) on
3 2
the(relative)charactervariety,andthebasictheoremofMarkoffcanbeinterpreted
assayingthatthesetofpositiveMarkofftriplesistheorbitoftheroottriple(3,3,3)
undertheactionofOut(F )(Cohn[4]wasthefirsttonoticetheconnectionbetween
2
the Markoff triples and traces of simple geodesics on the modular torus).
Various subsets of the character variety have interpretations as Fricke spaces
of geometric structures (for example the real points on (1) except the origin cor-
respond to hyperbolic structures on a once-punctured torus), and Out(F ) acts
2
properly discontinuously on these subsets. On the other hand, other subsets have
interpretations as representations of F into SU(2) and results of Goldman [5, 6]
2
imply that Out(F ) acts ergodically on these subsets. Hence, the overall action of
2
Γ∗ on C3 is dynamically interesting and quite mysterious.
3
Hurwitz [11] generalized the study to the n-variable diophantine equation
x2+x2+···+x2 −kx x ···x =0, (2)
1 2 n 1 2 n
(wherek ∈Z),andobtainedanalogousresultstothosefortheMarkoffequation. In
particular,heshowedthatthereareafinitesetofbasicsolutionsof(2)suchthatall
integral solutions can be obtained by considering the orbits of these basic solutions
under a certain group action. By a simple change of variables by a homothety
x (cid:55)→ λx, one may assume that k = 1 in the above, and consider the Markoff-
Hurwitz polynomial:
H(x ,x ··· ,x ):=x2+x2+···+x2 −x x ···x (3)
1 2 n 1 2 n 1 2 n
(n=3 corresponds to the Markoff polynomial on the left hand side of (1)).
In this context, it is interesting to determine the group of polynomial automor-
phismsofCn whichpreserves(3),andmoregenerallytostudythedynamicsofthis
groupactiononCn. Westateherethreegeneralquestionswhichweaddressinthis
paper:
Question 1. What is the group of polynomial automorphisms of Cn preserving
the Markoff-Hurwitz polynomial (3)?
Question 2. How can we describe invariant open subsets of Cn on which this
group acts properly discontinuously (domains of discontinuity)? These sets would
be analogues of Teichmu¨ller or Fricke spaces, and the quotient under the group
action the analogues of various moduli spaces.
Question3. Whatcanwesayabouttheorbitsofpointsinthedomainofdiscon-
tinuity, inparticular, whatistheirgrowthrateanddotheysatisfysomeidentities?
This question is motivated by results of McShane [14] who proved a remarkable
identity for once-punctured hyperbolic tori which can be interpreted as an iden-
tity satisfied by the Markoff triples, and more generally, triples of real numbers
satisfying (1). This was generalized by Bowditch [3], Tan-Wong-Zhang [21] and
Hu-Tan-Zhang [9] in the context of the group action on C3 preserving (1). The
natural question here is whether these identities generalize for Cn, n≥4.
The main purpose of this paper is to give answers to the above questions. For
Question 1, we have the following (compare to [7] which considers the case n=3):
4 H.HU,S.P.TAN,ANDY.ZHANG
Theorem1.1. ThegroupofpolynomialautomorphismsofCn preservingtheMarkoff-
Hurwitz polynomial
H(x)=H(x ,x ,··· ,x ):=x2+x2+···+x2 −x x ···x (4)
1 2 n 1 2 n 1 2 n
is given by
Γ∗ =Γ (cid:111)Λ
n n
where
(1) Λ is the group of linear automorphisms of (Cn,H(x)) and is given by Υ (cid:111)
n
S where Υ is the group of even sign change automorphisms and S the
n n n
symmetric group acting by permutations on the coordinates; and
(2) Γ = (cid:104)b ,··· ,b | b2 = Id,i = 1,··· ,n(cid:105) where b , i = 1,··· ,n is an
n 1 n i i
involution of Cn which fixes x for j (cid:54)= i and replaces x with the product
j i
of the other coordinates subtract x , that is,
i
(cid:16) (cid:89) (cid:17)
b (x ,··· ,x ,··· ,x ):= x ,··· ,x , x −x ,x ,··· ,x .
i 1 i n 1 i−1 j i i+1 n
j(cid:54)=i
In particular, Γ is a normal subgroup of finite index in Γ∗.
n n
For the second question, it is convenient, and sufficient, to consider the action
of Γ on Cn. From a dynamical point of view, this is equivalent to studying the
n
actionof Γ∗ on Cn, sinceΓ isa normal subgroupoffiniteindexin Γ∗. Thegroup
n n n
Γ is easier to handle as it is a Coxeter group generated by n involutions with no
n
other relations, and the Cayley graph ∆ of Γ is a regular (edge-labeled), rooted
n
n-valence tree. The vertex set V(∆) consists of the elements of Γ (with root the
n
identity element Id ∈ Γ ) and the edge set E(∆) is labeled by 1,2,··· ,n; two
n
vertices g,h ∈ V(∆) are connected by an i-edge if and only if g = h·b . For any
i
subtree ∆(cid:48) ⊂ ∆, denote by V(∆(cid:48)) and E(∆(cid:48)) the vertex set and edge set of ∆(cid:48)
respectively, note that E(∆(cid:48)) has a labeling induced from the labeling on E(∆).
An element a=(a ,··· ,a )∈Cn induces a map
1 n
Φ:=Φ :V(∆)→Cn, where Φ(g)=g−1(a),
a
which satisfies the following edge relations:
Definition 1.2 (Edge relations). Suppose that g,h ∈ V(∆) are connected by an
i-edge, and Φ(g)=(x ,··· ,x ,··· ,x ) and Φ(h)=(y ,··· ,y ,··· ,y ). Then
1 i n 1 i n
x = y , j (cid:54)=i, 1≤j ≤n
j j
x +y = x ···xˆ ···x =y ···yˆ ···y (5)
i i 1 i n 1 i n
   
y x
1 1
. .
 .  i  . 
 .   . 
   
 yi =Φ(h) Φ(g)= xi 
 ...  h h·bi =g  ... 
y x
n n
Figure 1. Edge relation
AUTOMORPHISMS PRESERVING THE MARKOFF-HURWITZ POLYNOMIAL 5
We call maps from V(∆) to Cn which satisfy the edge relations vector-valued
Hurwitz maps. The vector-valued Hurwitz map Φ describes the orbit of a under
Γ . Furthermore, because of the edge relations, Φ is determined by its value on
n
any vertex, in particular, the root; so the set of vector-valued Hurwitz maps can
be identified with Cn.
Wenextdescribeanimportantsetofgeometricobjects,thealternatinggeodesics
(or 2-regular subtrees of ∆), which will play an important role in describing the
domain of discontinuity, as well as the identities. These are bi-infinite geodesics
γ ⊂ ∆ consisting of alternating i, j-edges, where 1 ≤ i < j ≤ n. Denote by A the
setofalternatinggeodesicsandA thesetofalternatinggeodesicsconsistingofi-,
ij
(cid:70)
j-edges. Then A= A , where the union is taken over all distinct pairs i,j. The
ij
set A has some very interesting geometric properties which are explored in greater
detail in §2.4 and §2.15. For the purpose of studying the dynamics of the group
action, we first note that a vector-valued Hurwitz map Φ : V(∆) → Cn induces a
map φ:A→C as follows: Given an {i,j}-alternating geodesic γ, we note that for
all vertices g ∈ V(γ) and k (cid:54)= i,j, the k-th entries of Φ(g) are the same since by
the edge relations (5) only the ith and jth entries change when moving along the
edges of the alternating geodesic.
Definition 1.3. Suppose γ ∈ A and g ∈ V(γ) with Φ(g) = (x ,x ,··· ,x ).
ij 1 2 n
Then the (extended) Hurwitz map φ:A→C is given by
(cid:89)
φ(γ)= x .
k
k(cid:54)=i,j
There is also a secondary map, the square sum weight σ :A→C, given by
(cid:88)
σ(γ)= x2
k
k(cid:54)=i,j
which will be useful for the identities later.
Fix a ∈ Cn, let Φ : V(∆) → Cn be the induced vector-valued Hurwitz map,
φ:A→C the induced (extended) Hurwitz map and let K >0. Define
A (K)={γ ∈A | |φ(γ)|≤K}.
φ
Our second main theorem answering Question 2 is as follows:
Theorem 1.4. Let D ⊂Cn be the set of a∈Cn for which the induced (extended)
Hurwitz map φ:=φ :A→C satisfy the conditions
a
(i) φ(γ)(cid:54)∈[−2,2] for all γ ∈A;
(ii) the set A (K) is finite for some K >2.
φ
Then D is a non-empty open subset of Cn invariant under Γ and Γ acts properly
n n
discontinuously on D. Furthermore, 0∈Cn does not lie in the closure of D.
Although the sets D above are defined by rather simple conditions, they are
typically very complicated with fractal like boundary. Figure 2 shows computer
generated images of the diagonal slices of the sets (where a=(a,a,··· ,a)∈D) in
the case where n = 3 and 4 (points in the black part correspond to points in D).
Furthermore,inthen=3case,thediagonalsliceisverydifferentfromthediagonal
slice of the Schottky space, a set which is defined from geometric considerations,
see [19].
6 H.HU,S.P.TAN,ANDY.ZHANG
Figure 2. Diagonal slices of the set D for the cases n = 3 (left)
and n=4, Courtesy of Yasushi Yamashita.
An important property of the sets A (K) for K ≥ 2 is that they are edge-
φ
connected (see Proposition 2.3). That is, if γ,γ(cid:48) ∈ A (K), there exists a sequence
φ
γ ,··· ,γ ∈ A (K) such that γ = γ , γ(cid:48) = γ and γ and γ share an edge for
1 m φ 1 m i i+1
i=1,··· ,m−1. This makes it possible to write a computer program to search for
the set A (K) when it is finite, and hence determine the diagonal slice of D.
φ
Another important property of the maps φ induced by elements of D is that
the function log+|φ| = max{0,log|φ|} has Fibonacci growth (see §2.15 for the
definition of the Fibonacci function on A and Propositions 3.10 and 3.11 for the
precise statement). This allows us to deduce the absolute convergence of certain
sums over A, which are used in the proofs of the last set of results concerning
identitesfororbitsofelementsinD. Moreprecisely, wehavethefollowingtheorem
which answers Question 3.
Theorem 1.5. Suppose that a = (a ,··· ,a ) ∈ D ⊂ Cn as in Theorem 1.4.
1 n
Suppose further that
a2+a2+···+a2 −a a ···a =µ.
1 2 n 1 2 n
Then
(cid:88)
h (γ)=1
µ
γ∈A
where the series converges absolutely and
(cid:115)
(cid:18) (cid:19)
2µ 4
h (γ):=1− 1+ 1− ,
µ n(n−1)(σ(γ)−µ) (φ(γ))2
where φ,σ : A → C are the (extended) Hurwitz map and the square sum weight
given in Definition 1.3.
In particular, when µ=0 we have
(cid:88)
h(φ(γ))=1
γ∈A
AUTOMORPHISMS PRESERVING THE MARKOFF-HURWITZ POLYNOMIAL 7
where the sum converges absolutely and for z (cid:54)=0
(cid:114)
4
h(z)=1− 1− .
z2
Remarks:
(1) Thecasewheren=3,(a ,a ,a )∈R3\{(0,0,0)}andµ=0istheoriginal
1 2 3
McShaneidentity,thecasewhere(a ,a ,a )∈C3,φsatisfiestheconditions
1 2 3
of Theorem 1.4 and µ = 0 is Bowditch’s generalization, and when µ (cid:54)= 0,
this was done by the authors in [9].
(2) For the case where n = 4 and µ = 0 Huang and Norbury [10] found a
veryinterestinggeometricinterpretationasanidentityforsimpleone-sided
geodesics on the thrice-punctured projective plane.
(3) There are other ways in which the group Γ can act on Cn which are of
n
interestgeometrically. Anexampleofthiscanbefoundin[13]whichstudies
theactionofΓ onC3 arisingfromtheactionofthemappingclassgroupon
3
therelativecharactervarietyofafour-holedsphere, fixingthetracesofthe
four boundary components. Another interesting example can be obtained
by looking at the relative character variety of the four-holed sphere where
wefixthetracesofthreeinteriorsimpleclosedcurvestobezeroandlookat
the action of a different subgroup of Out(F ) on this, preserving the traces
3
of the interior curves. In this case we get an action of Γ on C4 which is
4
similar to the one we consider. In these cases, the polynomials preserved
by the group action have degree n and the methods we use here can be
applied to these cases directly.
(4) Another very interesting example of actions of Γ on Rn occurs in the
n
studyofApolloniancircle/spherepackings,herethegroupactionpreserves
a quadratic polynomial and it would be interesting to try to apply the
techniques developed here to this class of problems.
(5) An earlier version of some of the results here are contained in the first
author’s Ph.D. thesis [8].
The rest of this paper is organized as follows. We defer the proof of Theorem
1.1 to the Appendix as the proof uses elementary techniques which are somewhat
independentfromtherestofthepaper. In§2,wedescribethecombinatorialsetting
for the problem, set up the notation, introduce some of the basic geometric and
combinatorial objects, and prove some basic results. §3 is one of the main parts of
thepaperwhereweprovesomeofthemainresultsforHurwitzmaps,leadingupto
the proof of Theorem 1.4. In §4 we discuss the identities and prove Theorem 1.5.
Finally, in §5, we discuss some open problems and directions for further study.
Acknowledgements. We are grateful to Martin Bridgeman, Dick Canary, Fran¸cois
Labourie, Bill Goldman, Makoto Sakuma and Yasushi Yamashita for helpful con-
versations and comments. In particular, we are very grateful to Dick Canary who
had asked about the domain of continuity for the action of Γ on Cn, to Yasushi
n
Yamashita who had written the computer program which drew the beautiful slices
in Figure 2, and to Bill Goldman for his constant support and encouragement. We
would also like to thank Yi Huang and Paul Norbury for bringing our attention to
[10] and the geometric interpretation of our identity for the n = 4 homogeneous
case, see Remark (2) above.
8 H.HU,S.P.TAN,ANDY.ZHANG
2. Combinatorial setting and basic results
2.1. Cayley graphs and orbits. Let Γ :=(cid:104)b ,··· ,b |b2 =···=b2 =Id(cid:105) act
n 1 n 1 n
on Cn as follows:
(cid:16) (cid:89) (cid:17)
b (x ,··· ,x ,··· ,x ):= x ,··· ,x , x −x ,x ,··· ,x .
i 1 i n 1 i−1 j i i+1 n
j(cid:54)=i
Clearly, the Markoff-Hurwitz polynomial (4) is invariant under Γ .
n
The Cayley graph ∆ := ∆(Γ ) of Γ is a rooted, regular n-tree with vertex set
n n
V(∆) = Γ , where the root v corresponds to the identity element of Γ. Two
n 0
vertices g,h ∈ V(∆) are connected by an edge if and only if g = hb for some
i
i ∈ {1,2,··· ,n}. The corresponding edge of ∆ is labeled (or colored) by i, so the
edgesincidenttoanygivenvertexarelabeledbythedistinctelementsof{1,··· ,n},
giving a regularly labeled n-tree. See Figure 3 for the case of n=4.
1
2
3
4
v
0
Figure 3. The rooted regular tree ∆ for n=4
An element a∈Cn induces a vector-valued Hurwitz map
Φ:=Φ :V(∆)→Cn
a
where Φ(g) = g−1(a). Φ satisfies the edge relations (5). Furthermore, the set
{x=Φ(g) | g ∈Γ } is the orbit of a under Γ .
n n
There are many interesting combinatorial objects and structures associated to
∆,andmapsfromtheseobjectstoCinducedfromΦwhichwedescribeinthenext
sections. We will emphasize the combinatorial setting. The group action will be
reflected by various edge relations satisfied by the maps.
2.2. Index sets [n], P(n), Iˆ, {ˆi} and C(n,k). We set some notation for index
sets. Let [n] := {1,2,··· ,n} and P(n) the power set of [n]. For I ∈ P(n), |I|
denotesthecardinalityofI andIˆ:=[n]\I denotesthecomplementofI in[n]. Let
C(n,k)={I ∈P(n) : |I|=k}. For I ={i}, use {ˆi} to denote Iˆ.
AUTOMORPHISMS PRESERVING THE MARKOFF-HURWITZ POLYNOMIAL 9
2.3. Rooted regular trees. Let ∆ be a rooted, regularly labeled n-tree, that is,
ann-valenttreewithadistinguishedvertexv , theroot, suchthatincidenttoeach
0
vertex, the n edges are labeled by distinct elements of [n].
We put the standard metric on ∆ where every edge has length one and denote
by d(T) the distance of any subtree T ⊂∆ to v .
0
DenotebyV(∆)andE(∆)thevertexsetandtheedgesetof∆respectively. For
e ∈ E(∆) denote by e¯the closure of e, that is, the subtree consisting of e and the
two vertices incident to e. Every e∈E(∆) can be directed in two ways, denote by
E(cid:126)(∆) the set of directed edges of ∆ and for (cid:126)e ∈ E(cid:126)(∆) denote by −(cid:126)e the directed
edge with the same underlying edge e but directed in the opposite direction of (cid:126)e.
We think of a directed edge (cid:126)e as specifying the two vertices incident to e as the
tail and head (we speak of(cid:126)e as being ‘directed towards’ its head). Note that every
directed edge(cid:126)e is either directed towards the root v , or away from it.
0
2.4. Regular subtrees: T|k|(∆), X := T|n−1|(∆) and A := T|2|(∆). Let ∆ be
a rooted regularly labeled n-tree and let I ∈ C(n,k). Denote by E (∆) ⊂ E(∆)
I
the set of edges labeled by the elements of I; in particular E(∆) = (cid:70)n E (∆).
i=1 {i}
Then ∆\E (∆) is a disjoint union of I-regular trees, we call the set of connected
Iˆ
componentsof∆\E theI-forestof∆,denotedbyTI(∆); itselementsareexactly
Iˆ
theregularlylabeledk-subtreesof∆labeledbyI. Wedenotethesetofallregularly
labeled k-subtrees of ∆ by T|k|(∆), so
(cid:71)
T|k|(∆)= TI(∆).
I∈C(n,k)
In particular, we have the following decomposition of T|n−1|(∆):
n
T|n−1|(∆)= (cid:71)T{ˆi}(∆).
i=1
Thus T|0|(∆)=V(∆), T|1|(∆)=E(∆):={e¯| e∈E(∆)}, T|2|(∆) is the set of
alternatinggeodesicsintroducedin§1,andT|n|(∆)={∆}. Weshallbeparticularly
interested in T|2|(∆) and T{i,j}(∆), 1≤i<j ≤n, which we denote by A and A ,
ij
thesearethesetofalternatinggeodesicsdescribedintheintroduction. Inaddition,
T|n−1|(∆) is denoted by X, and T{ˆi}(∆) by X . See Figure 4 for examples of the
i
case n=4. We will be mostly interested in maps from X and A into C induced by
a vector-valued Hurwitz map Φ.
Aregularlylabeledk-subtreeof∆isdeterminedbyitscolorsetI ∈C(n,k)and
any one of its vertices, v, hence we can denote it by
[v;I]∈TI(∆).
The vertex v is not unique, nonetheless, there is a unique vertex v∗ ∈ V([v;I])
which is closest to the root v ∈V(∆) so d([v;I])=d(v∗)=l(α) where l(α) is the
0
length of the geodesic α from v to v∗. Hence, [v;I] inherits a rooted structure,
0
with root v∗.
For convenience, we adopt the following conventions. Elements of A will be
denoted with Greek letters α,β,γ,δ. Elements of X will be denoted by upper case
letters X,Y or Z.
10 H.HU,S.P.TAN,ANDY.ZHANG
v v
Figure 4. Examples of an element in A (left) and an element in
X for the case n=4
Suppose that I ∈C(n,k). Then
(cid:92)
[v;I]= [v;{ˆj}].
j∈Iˆ
In particular,
n n
(cid:92) (cid:92)
v =[v;∅]= [v;{ˆj}]= X ,
j
j=1 j=1
where X :=[v;{ˆj}]∈X .
j j
Let e ∈ E (∆) and V(e¯) = {v,v(cid:48)}, that is, e is labeled by i and v and v(cid:48) are
{i}
the vertices incident to e. For j ∈ [n], let X := [v;{ˆj}] and X(cid:48) := [v(cid:48);{ˆj}]. Then
j j
X =X(cid:48) for j (cid:54)=i, but X (cid:54)=X(cid:48). In this case, we write
j j i i
e↔({X ,··· ,Xˆ ,··· ,X };{X ,X(cid:48)}).
1 i n i i
We shall also write
(cid:126)e↔({X ,··· ,Xˆ ,··· ,X };X →X(cid:48))
1 i n i i
for the directed edge(cid:126)e on e pointing away from X and towards X(cid:48).
i i
2.5. Hurwitz maps. We define functions from X =T|n−1|(∆) to C induced from
the vector-valued Hurwitz functions Φ defined in §2.1.
Definition 2.1. A Hurwitz map is a function φ : X → C satisfying the following
edge relations: for each e∈E (∆), if e↔({X ,··· ,Xˆ ,··· ,X };{X ,X(cid:48)}) then
{i} 1 i n i i
(cid:89)
φ(X )+φ(X(cid:48))= φ(X ). (6)
i i j
j(cid:54)=i
Convention: ForafixedHurwitzmapφandforeachX ∈X,usethecorresponding
lowercaselettertodenoteφ(X),forexample,x =φ(X ),y =φ(Y). Wewilladopt
i i
this convention in the rest of this paper.
Equation (6) above can then be written as
(cid:89)
x +x(cid:48) = x . (7)
i i j
j(cid:54)=i