Reconstruction of invariants of configuration spaces of hyperbolic curves from associated Lie algebras

Reconstruction of invariants of configuration spaces of hyperbolic curves from associated Lie algebras

Koichiro Sawada

Research Institute for Mathematical Sciences, Kyoto University

Combinatorial Anabelian Geometry and Related Topics 2021/07/06

Table of contents

§1 Introduction

§2 Reconstruction algorithms

§ 1 Introduction

K: algebraically closed field of characteristic zero X: hyperbolic curve/K of type (g, r)

X_n:={(x₁, . . . , x_n) ∈ X×^K· · ·×^KX | x_i ̸= x_j (∀i ̸= j)} : n-th configuration space of X

Π^Σ_n := π₁(X_n)^Σ: maximal pro-Σ quotient of π₁(X_n) (Σ = {l} for ^∃l or Σ: the set of all prime numbers) We shall refer to a profinite group isom to

Π^Σ₁ (for some X) as a surface group.

Theorem A (Hoshi-Minamide-Mochizuki (preprint,2017)) Π^Σ_n ^recon.⇝ n, g, r,“generalized fiber subgroups”

if n ≥2

Theorem B (S. (preprint,2018))

Gr(Π^l_n) ^recon.⇝ n, g, r,“generalized fiber ideals”

if n ≥2

(More precisely, there are algorithms of reconstructing these invariants from the abstract Lie algebra over Z^l obtained by forgetting the grading of Gr(Π^l_n).)

My research on Theorem B was motivated by [HMM].

Today, we will compare their algorithms to observe

similarities (i.e., what can be applied commonly to both) and differences.

Definition (generalized projection) p :X_n →X_m (0≤ m ≤ n)

is a generalized projection morphism ^def⇔ If (g, r) ∈ {/ (0,3),(1,1)}, then p : Xn →Xm: projection morphism If (g, r) = (0,3), then Xn ∼= (M^0,n+3)K

p : X_n →^∼ (M^0,n+3)_K → (M^0,m+3)_K →^∼ X_m

If (g, r) = (1,1), then X_n ∼= E_n+1/E (E := X^cpt) p : X_n →^∼ E_n+1/E → E_m+1/E →^∼ X_m

Definition (generalized fiber subgroup)

A generalized projection morphism induces Π^Σ_n ↠ Π^Σ_m. ker(Π^Σ_n ↠Π^Σ_m): generalized fiber subgroup

(of co-length m)

GFS_m(Π^Σ_n): the set of gen. fiber subgps of co-length m

Note: N ∈ GFSm(Π^Σ_n) is isomorphic to “Π^Σ_n−m” of a hyperbolic curve of type (g, r+m).

Definition (Lie algebra associated to X_n) Write Π^l_n(1) := Π^l_n(:= Π^{n^l}),

Π^l_n(2):= ker(Π^l_n ↠ (π₁(X^cpt×^K · · · ×^K X^cpt)^l)^ab), Π^l_n(m) := ⟨[Π^l_n(m1),Π^l_n(m2)] | m1 +m2 = m⟩ (m ≥ 3), Gr^m(Π^l_n) := Π^l_n(m)/Π^l_n(m + 1),

Gr(Π^l_n) := ⊕

m≥1Gr^m(Π^l_n).

Then Gr(Π^l_n): graded Lie algebra over Z^l.

We shall refer to an abstract Lie algebra/Zl isomorphic to Gr(Π^l₁) as a surface algebra.

Definition (generalized fiber ideal)

A (generalized) projection morphism X_n → X_m induces Gr(Π^l_n) ↠Gr(Π^l_m).

ker(Gr(Π^l_n) ↠Gr(Π^l_m)): (generalized) fiber ideal (of co-length m)

GFI_m(Gr(Π^l_n)): the set of gen. fib. ideals of co-length m

Note: i ∈ GFI_m(Gr(Π^l_n)) is isomorphic to “Gr(Π^l_n−m)”

of a hyperbolic curve of type (g, r+m).

Gr(Π^l_n) has a presentation with generators X_i^(k), Y_i^(k) ∈ Gr¹(Π^l_n), Z_j^(k), W_h^(k) ∈ Gr²(Π^l_n) (1 ≤i ≤g, 1≤ j ≤ r, 1≤ k, h ≤ n)

and relations (R1–10):

∑g i=1

[X_i^(k), Y_i^(k)] +

∑r j=1

Z_j^(k)+

∑n h=1

W_h^(k) = 0, (R1)

W_k^(k) = 0, (R2)

W_h^(k) = W_k^(h), (R3)

[X_i^(k), X_i^(k_′ ^′)] = [Y_i^(k), Y_i^(k_′ ^′)] = 0 (k ̸= k^′), (R4) [X_i^(k), Y_i^(k_′ ^′)] = 0 (i ̸= i^′, k ̸= k^′), (R5) [X_i^(k), Y_i^(k′)] = W_k^(k′) (k ̸= k^′), (R6) [X_i^(k), Z_j^(k′)] = [Y_i^(k), Z_j^(k′)] = 0 (k ̸= k^′), (R7) [Z_j^(k), Z_j^(k_′^′)] = 0 (j ̸= j^′, k ̸= k^′), (R8) [X_i^(k), W_h^(k′)] = [Y_i^(k), W_h^(k′)] = [Z_j^(k), W_h^(k′)] = 0

(k /∈ {k^′, h}), (R9) [W_h^(k), W_h^(k_′^′)] = 0 ({k, h} ∩ {k^′, h^′} = ∅). (R10)

X₁⁽¹⁾· · · Xg⁽¹⁾Y₁⁽¹⁾· · · Yg⁽¹⁾Z₁⁽¹⁾· · · Zr⁽¹⁾ 0 W₂⁽¹⁾· · · Wn⁽¹⁾

X₁⁽²⁾· · · Xg⁽²⁾Y₁⁽²⁾· · · Yg⁽²⁾Z₁⁽²⁾· · · Zr⁽²⁾W₁⁽²⁾ 0 · · · Wn⁽²⁾

... ... ... ... ... ... ... ... ... ...

X₁⁽ⁿ⁾· · · Xg⁽ⁿ⁾Y₁⁽ⁿ⁾· · · Yg⁽ⁿ⁾Z₁⁽ⁿ⁾· · · Zr⁽ⁿ⁾W₁⁽ⁿ⁾W₂⁽ⁿ⁾· · · 0

§ 2 Reconstruction algorithms

・Π^Σ_n ^recon.⇝ n

Proposition 1

Let H ⊂Π^l_n: closed subgroup isom to Z^⊕_l ^s. Then s ≤ n.

(Proof) n= 1: a property of pro-l surface groups n ≥2: consider 1 →Π^l_n/n−1 → Π^l_n → Π^l_n−1 → 1 (Π^l_n/n−1: surface group)

Theorem 2

∃H ⊂ Π^l_n: closed subgroup isom to Z^⊕_l ⁿ.

(This follows from the existence of log-full points of the log configuration space of a suitable stable log curve.) Algorithm:

Take l: a prime number s.t. (Π^Σ_n)^l ̸= {1} (⇔l ∈ Σ) Π^l_n = (Π^Σ_n)^l

n= max{s∈Z≥0|^∃H: closed subgp of Π^l_n s.t. H ∼= Z^⊕l ^s}

・Gr(Π^l_n) ^recon.⇝ n

Proposition 3

Let a ⊂ Gr(Π^l_n): abelian subalgebra/Z^l isom to Z^⊕_l ^s as a Z^l-module. Then s ≤n.

(Lie alg. L is abelian ^def⇔ ^∀a, b ∈ L [a, b] = 0)

(Proof) n= 1: [a, b] = 0⇒ a, b: linearly dependent n ≥2: consider

1 →Gr(Π^l_n/n−1) → Gr(Π^l_n) → Gr(Π^l_n−1) →1

Theorem 4

∃a ⊂ Gr(Π^l_n): abelian subalg. isom to Z^⊕l ⁿ. (Proof)

If g > 0, then [X₁^(h), X₁^(k)] = 0 for 1 ≤h < k ≤ n.

If r > 0, then write A^(k) := Z_r^(k)+

k−1

∑

h=1

W_h^(k). Then [A^(h), A^(k)] = 0 for 1 ≤h < k ≤n.

Algorithm:

n = max{s ∈ Z≥0|^∃a : abelian subalg. of Gr(Π^l_n) s.t. a ∼= Z^⊕s}

・Π^Σ_n ^recon.⇝ GFS₁(Π^Σ_n)

Theorem 5

Let H ⊂Π^Σ_n: normal closed subgroup such that Π^Σ_n/H is isom to a surface group which is not free of rank 2.

Then ^∃N ∈ GFS₁(Π^Σ_n) s.t. N ⊂ H.

In particular, (g, r) ∈ {/ (0,3),(1,1)} ⇔ ^∃H as above.

Proposition 6

Π^l,ab_n : free Z^l-module,

rank_ZlΠ^l,ab_n =









n(r −1) +n(n−1)/2 (g = 0)

2gn (r = 0)

n(2g+r −1) (g, r >0).

In particular, (g, r) = (1,1)⇔ rank_ZlΠ^l,ab_n = 2n.

Algorithm:

S := {H ^cl.◁ Π^Σ_n|Π^Σ_n/H : surf. gp, not free of rank 2} If S ̸= ∅ (⇔(g, r) ∈ {/ (0,3),(1,1)}), then

GFS₁(Π^Σ_n): minimal elements of S.

S = ∅ · · · we omit the details

((g, r) = (1,1)⇔ rank_ZlΠ^l,ab_n = 2n,

construct Lie alg. isom to Gr(Π^l_n) (∼= Gr^lcs(Π^l_n)), consider Π^Σ_n → Π^l,ab_n ⊗Zl Q^l,. . .)

・Gr(Π^l_n) ^recon.⇝ GFI₁(Gr(Π^l_n)) Theorem 7

We can classify all (not necessarily graded) surjective homomorphisms/Z^l from Gr(Π^l_n) to a surface

algebra/Z^l.

(Proof: direct calculation)

Corollary 8

Let i ⊂ Gr(Π^l_n): an ideal/Z^l (not necessarily graded).

Then i ∈ GFI₁(Gr(Π^l_n)) ⇔ Gr(Π^l_n)/i ∼= Gr(Π^l₁).

Corollary 9

Gr(Π^l₁): unique surface algebra s (up to isom) s.t.

• Gr(Π^l_n) ↠^∃ s,

• for any surface alg. h, if Gr(Π^l_n) ↠^∃ h, then s ↠^∃ h.

Algorithm:

(isom class of) Gr(Π^l₁):

(unique) surface algebra s satisfying

• Gr(Π^l_n) ↠^∃ s,

• for any surface alg. h, if Gr(Π^l_n) ↠^∃ h, then s ↠^∃ h.

GFI1(Gr(Π^l_n)) = {i

ideal/Zl

⊂ Gr(Π^l_n)|Gr(Π^l_n)/i∼= Gr(Π^l₁)}

・Π^Σ_n ^recon.⇝ GFS_m(Π^Σ_n), Gr(Π^l_n) ^recon.⇝ GFI_m(Gr(Π^l_n))

Algorithm:

GFS_m(Π^Σ_n) = ∪

H∈GFSm−1(Π^Σ_n)

GFS₁(H)

GFI_m(Gr(Π^l_n)) = ∪

i∈GFI_m−1(Gr(Π^l_n))

GFI₁(i)

(We set GFS₁({1}) := ∅, GFI₁({0}) := ∅.)

Remark

・We do not use “n” for Gr(Π^l_n) ^recon.⇝ GFI_m(Gr(Π^l_n)).

So we can obtain other algorithms for Gr(Π^l_n) ^recon.⇝ n.

For example, n: unique nonnegative integer m s.t.

GFI_m(Gr(Π^l_n)) = { {0}}

.

・We can reconstruct GFS₁(Π^Σ_n) from Π^Σ_n in a similar manner to Gr(Π^l_n) ^recon.⇝ GFI₁(Gr(Π^l_n)).

・Π^Σ_n ^recon.⇝ (g, r) (if n≥ 2) Proposition 10

・rank_ZlΠ^l,ab₁ = 2g + max{r −1,0}

・Π^l₁: free ⇔ r > 0

Theorem 11 ([CbTpI] Lemma 1.3(iv)) Suppose: r > 0.

Consider the natural action Π^l₁ →Aut(Π^l,ab_2/1) determined by 1→ Π^l_2/1 →Π^l₂ → Π^l₁ →1.

Then ker(Π^l₁ →Aut(Π^l,ab_2/1)) = ker(Π^l₁ ↠ π₁(X^cpt)^l,ab).

Algorithm(1/2): (Suppose: n ≥2.) Take H ∈ GFS_n−2(Π^l_n), N ∈ GFS₁(H).

(Π^l₂ ∼= Π^l_n/H, Π^l₁ ∼= H/N)

If H/N: not free (⇔ r = 0), then g = 1

2rank_Zl(H/N)^ab, r = 0.

Algorithm(2/2):

If H/N: free, then, since

ker(Π^l₁ ↠ π₁(X^cpt)^l,ab) = ker(H/N →Aut(N^ab)), ker(Π^l,ab₁ ↠π₁(X^cpt)^l,ab)

= Im(ker(H/N →Aut(N^ab)) →(H/N)^ab).

r = rank_Zl(ker(Π^l,ab₁ ↠ π₁(X^cpt)^l,ab)) + 1 g = 1

2(rank_Zl(H/N)^ab −r + 1)

・Gr(Π^l_n) ^recon.⇝ (g, r) Write Gr(Π^l_n)(2) := ⊕

m≥2

Gr^m(Π^l_n),

Gr(Π^l_n)[1] := Gr(Π^l_n),

Gr(Π^l_n)[m] := [Gr(Π^l_n),Gr(Π^l_n)[m −1]].

Theorem 12

Suppose: n ≥2 and g > 0.

Let i,j: distinct elements of GFIn−1(Gr(Π^l_n)).

Then i∩Gr(Π^l_n)(2) = {a ∈ i|^∀b ∈ j [a, b]∈Gr(Π^l_n)[3]}.

(Proof: if j ̸= j^′ and k ̸= k^′,

[Z_j^(k), X_i^(k′)] = [Z_j^(k), Y_i^(k′)] = [Z_j^(k), Z_j^(k_′ ^′)] = 0, [Z_j^(k), W_h^(k′)] = [Z_j^(k),[X₁^(h), Y₁^(k′)]] ∈ Gr(Π^l_n)[3], [Z_j^(k), Z_j^(k′)] = −[Z_j^(k), W_k^(k′)] ∈ Gr(Π^l_n)[3].)

Proposition 13

・rank_Zl(Gr(Π^l₁)/Gr(Π^l₁)[2]) = 2g+ max{r −1,0}

・Gr(Π^l₁): free ⇔ r > 0

It suffices to determine whether g = 0 or not.

Idea: If g = 0, there are many homomorphisms determined as follows:

X ,→ Y: of type (0,3),

p :Y_n → Y: a gen. proj. which is not a proj., Gr(Π^l_n) ↠Gr(π1(Y)^l): determined by Xn ,→Yn

→p Y. (exceptional mor.)

Theorem 14

Write E := {i ^ideal/⊂^Zl Gr(Π^l_n)|Gr(Π^l_n)/i : free of rank 2,

∀j ∈ GFI₁(Gr(Π^l_n)) j ̸⊂ i}.

Then ♯E =







 (_r

2

)·(_n

2

)+r ·(_n

3

)+ (_n

4

) (g = 0, r ̸= 3) (_n

2

) (g = 1, r ̸= 1)

0 (else),

♯GFI1(Gr(Π^l_n)) =







 (_n+3

4

) ((g, r) = (0,3)) (_n+1

2

) ((g, r) = (1,1)) n (else).

Algorithm(1/2): (Suppose: n ≥2.) Write E := {i

ideal/Zl

⊂ Gr(Π^l_n)|Gr(Π^l_n)/i : free of rank 2,

∀j ∈ GFI₁(Gr(Π^l_n)) j ̸⊂ i}. (Since free Lie algebra/Z^l of rank 2 is a surface algebra, we can reconstruct E by Theorem 7.)

If ♯E ≤ (_n

2

) and ♯GFI₁(Gr(Π^l_n)) < (_n+3

4

) (⇔g > 0),

then take i,j: distinct elements of GFI_n−1(Gr(Π^l_n)).

i∩Gr(Π^l_n)(2) = {a ∈ i|^∀b ∈ j [a, b]∈Gr(Π^l_n)[3]} g = 1

2rank_Zl(i/i∩Gr(Π^l_n)(2))

Algorithm(2/2):

If ♯E > (_n

2

) or ♯GFI₁(Gr(Π^l_n)) = (_n+3

4

), then g = 0.

If Gr(Π^l₁): free (⇔ r > 0), then

r = rank_Zl(Gr(Π^l₁)/Gr(Π^l₁)[2]) + 1−2g.

If Gr(Π^l₁): not free, then r = 0.