Monodromy representations of meromorphic projective structures

Gupta, Subhojoy; Mj, Mahan

doi:10.1090/proc/14866

Cited by 6 publications

(12 citation statements)

References 6 publications

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“…This admits a monodromy map

F

to the corresponding space of framed representations

{\overset{χ}{true}}_{g} false(frakturn false)

. By combining the techniques in [17] and this paper, we deduce Theorem The image of the monodromy map

F : {scriptP}_{g}^{*} false(frakturn false) \to {\overset{χ}{true}}_{g} false(frakturn false)

is precisely the subset of non‐degenerate framed representations that have non‐trivial monodromy around the punctures corresponding to the poles of order at most two.…”

Section: Introductionmentioning

confidence: 69%

“…Indeed, the framing determined by

normalΨ

depends only on the leaves of

λ

of finite weight (c.f. [17, § 3.2.3]). To see this, let

γ

be a geodesic boundary component of

\overset{S}{true}

; this is a leaf of

λ

of infinite weight (see the proof of Proposition 3.3).…”

Section: Proofs Of Theorems 11–14mentioning

confidence: 99%

“…Our proof, that we will now describe, is a straightforward combination of techniques of [17] with those in the preceding sections. As described in [17, § 2.1 and § 2.2] (see also [1]), the

C {normalP}^{1}

‐structures in

{scriptP}_{g}^{*} false(frakturn false)

can be thought of having an underlying marked and bordered surface

S_{g} false(frakturn false)

. This can be equipped with a hyperbolic metric such that the points in

double-struckP

correspond to cusps, and the punctures in

P^{'}

correspond to geodesic boundary components with distinguished points on it (the number of which is two less than the corresponding entry of

frakturn

).…”

Section: Proofs Of Theorems 11–14mentioning

confidence: 99%

“…Equivalently, in our terminology,

\overset{ρ}{true}

does not satisfy conditions (a) and (b) of Definition 2.6, where

double-struckP

is the set of punctures taken into consideration for (b), and neither does the following additional condition hold: (3) There is some boundary arc

I

and some lift

\overset{I}{true}

in the universal cover whose endpoints in

F_{\infty}

are assigned the same point in

C {normalP}^{1}

by the framing

β

. (See

(D 1)

in [17, § 2.4.2] and (D1) of [1, Definition 4.3].) Moreover, we will assume that

\overset{ρ}{true}

has non‐trivial monodromy around each puncture in

double-struckP

.…”

Section: Proofs Of Theorems 11–14mentioning

confidence: 99%

“…Moreover, we will assume that

\overset{ρ}{true}

has non‐trivial monodromy around each puncture in

double-struckP

. Recall from [17, § 2.5] that an ideal triangulation

T

S_{g} false(frakturn false)

has vertices at

double-struckP

and at the distinguished points on each geodesic boundary component; moreover, a signing is a map

ε : P \to {- 1, 1}

(see [1, § 9.3]). By [1, Theorem 9.1] there is a signed triangulation

(T, ε)

such that the Fock–Goncharov coordinates (or cross‐ratio coordinates) of

\overset{ρ}{true}

with respect to

(T, ε)

are well defined and non‐zero.…”

Section: Proofs Of Theorems 11–14mentioning

confidence: 99%

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Monodromy groups of CP1‐structures on punctured surfaces

Gupta

2021

Journal of Topology

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For a punctured surface S, we characterize the representations of its fundamental group into normalPSL2false(double-struckCfalse) that arise as the monodromy of a meromorphic projective structure on S with poles of order at most two and no apparent singularities. This proves the analogue of a theorem of Gallo–Kapovich–Marden concerning CnormalP1‐structures on closed surfaces, and settles a long‐standing question about characterizing monodromy groups for the Schwarzian equation on punctured spheres. The proof involves a geometric interpretation of the Fock–Goncharov coordinates of the moduli space of framed normalPSL2false(double-struckCfalse)‐representations, following ideas of Thurston and some recent results of Allegretti–Bridgeland.

show abstract

“…This admits a monodromy map

F

to the corresponding space of framed representations

{\overset{χ}{true}}_{g} false(frakturn false)

. By combining the techniques in [17] and this paper, we deduce Theorem The image of the monodromy map

F : {scriptP}_{g}^{*} false(frakturn false) \to {\overset{χ}{true}}_{g} false(frakturn false)

is precisely the subset of non‐degenerate framed representations that have non‐trivial monodromy around the punctures corresponding to the poles of order at most two.…”

Section: Introductionmentioning

confidence: 69%

“…Indeed, the framing determined by

normalΨ

depends only on the leaves of

λ

of finite weight (c.f. [17, § 3.2.3]). To see this, let

γ

be a geodesic boundary component of

\overset{S}{true}

; this is a leaf of

λ

of infinite weight (see the proof of Proposition 3.3).…”

Section: Proofs Of Theorems 11–14mentioning

confidence: 99%

“…Our proof, that we will now describe, is a straightforward combination of techniques of [17] with those in the preceding sections. As described in [17, § 2.1 and § 2.2] (see also [1]), the

C {normalP}^{1}

‐structures in

{scriptP}_{g}^{*} false(frakturn false)

can be thought of having an underlying marked and bordered surface

S_{g} false(frakturn false)

. This can be equipped with a hyperbolic metric such that the points in

double-struckP

correspond to cusps, and the punctures in

P^{'}

correspond to geodesic boundary components with distinguished points on it (the number of which is two less than the corresponding entry of

frakturn

).…”

Section: Proofs Of Theorems 11–14mentioning

confidence: 99%

“…Equivalently, in our terminology,

\overset{ρ}{true}

does not satisfy conditions (a) and (b) of Definition 2.6, where

double-struckP

is the set of punctures taken into consideration for (b), and neither does the following additional condition hold: (3) There is some boundary arc

I

and some lift

\overset{I}{true}

in the universal cover whose endpoints in

F_{\infty}

are assigned the same point in

C {normalP}^{1}

by the framing

β

. (See

(D 1)

in [17, § 2.4.2] and (D1) of [1, Definition 4.3].) Moreover, we will assume that

\overset{ρ}{true}

has non‐trivial monodromy around each puncture in

double-struckP

.…”

Section: Proofs Of Theorems 11–14mentioning

confidence: 99%

“…Moreover, we will assume that

\overset{ρ}{true}

has non‐trivial monodromy around each puncture in

double-struckP

. Recall from [17, § 2.5] that an ideal triangulation

T

S_{g} false(frakturn false)

has vertices at

double-struckP

and at the distinguished points on each geodesic boundary component; moreover, a signing is a map

ε : P \to {- 1, 1}

(see [1, § 9.3]). By [1, Theorem 9.1] there is a signed triangulation

(T, ε)

such that the Fock–Goncharov coordinates (or cross‐ratio coordinates) of

\overset{ρ}{true}

with respect to

(T, ε)

are well defined and non‐zero.…”

Section: Proofs Of Theorems 11–14mentioning

confidence: 99%

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Monodromy groups of CP1‐structures on punctured surfaces

Gupta

2021

Journal of Topology

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Dominating surface-group representations into $${\textrm{PSL}_2 (\mathbb {C})}$$ in the relative representation variety

Gupta

Su²

2022

manuscripta math.

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Tame and relatively elliptic ℂℙ1–structures on the thrice-punctured sphere

Ballas,

Bowers,

Casella

et al. 2024

Algebr. Geom. Topol.

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Monodromy representations of meromorphic projective structures

Cited by 6 publications

References 6 publications

Monodromy groups of CP1‐structures on punctured surfaces

Monodromy groups of CP1‐structures on punctured surfaces

Dominating surface-group representations into $${\textrm{PSL}_2 (\mathbb {C})}$$ in the relative representation variety

Tame and relatively elliptic ℂℙ1–structures on the thrice-punctured sphere

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