Quotients of surface groups and homology of finite covers via quantum representations

Koberda, Thomas; Santharoubane, Ramanujan

doi:10.1007/s00222-016-0652-x

Cited by 34 publications

(48 citation statements)

References 23 publications

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“…Finally, in order to prove (3), observe that

ϕ false(π_{1} (S_{g^{'}}) false) = π_{1} false(S_{g^{'}} false)

, by construction. In particular,

ϕ (π_{1} false(S_{g} false))

contains

π_{1} false(S_{g^{'}} false)

as a subgroup of finite index and we can again apply the result of Koberda–Santharoubane [, Theorem 4.1] used above to deduce that (3) holds. At this point, Lemma tells us that the group

Q_{1} : = false⟨ ⟨ {prefixproj}_{g}^{p} false(h^{p} false) ⟩ false⟩

is an infinite normal subgroup of infinite index of

{prefixMod}_{g}^{1} / {prefixMod}_{g}^{1} false[p false]

.…”

Section: Normal Subgroups Via Coversmentioning

confidence: 74%

“…We first explain how to construct the group

Q_{1}

from the statement. First, there exists

h_{1} \in π_{1} false(S_{g} false)

of infinite order in

{prefixMod}_{g}^{1} / {prefixMod}_{g}^{1} false[p false]

: this is a consequence of work by Koberda–Santharoubane [, Theorem 4.1] for large

p

, and Funar–Lochak [, Proof of Proposition 3.2] for all

p

as in the hypotheses of Theorem . We are going to produce an injective homomorphism

{prefixMod}_{g}^{1} \to {prefixMod}_{g^{'}}^{1}

as in Lemma , suited to this element

h_{1}

.…”

Section: Normal Subgroups Via Coversmentioning

confidence: 99%

“…Then the subgroups

{\bar{Q}}_{i} = Q_{i} \cap \frac{π_{1} false(S_{g} false)}{π_{1} false(S_{g} false) false[p false]}

are normal subgroups of

\frac{π_{1} (S_{g})}{π_{1} (S_{g}) [p]}

thus forming a descending normal series. Each

{\bar{Q}}_{i}

is infinite and of infinite index into

{\bar{Q}}_{i - 1}

by the proof above, based on the Koberda–Santharoubane infiniteness statement [, Theorem 1.1] . We have:

false⟨ ⟨ {prefixproj}_{g}^{p} false(h_{1}^{p} false), \dots, {prefixproj}_{g}^{p} false(h_{j}^{p} false) ⟩ false⟩ \subset \frac{π_{1} false(S_{g} false)}{π_{1} false(S_{g} false) false[p false]} .

In fact, the groups

⟨ false⟨ {proj}_{g}^{p} (h_{1}^{p}), \dots, {proj}_{g}^{p} (h_{j}^{p}) false⟩ ⟩

are characteristic subgroups of

\frac{π_{1} (S_{g})}{π_{1} (S_{g}) [p]}

, and in particular they are also normal subgroups in

\frac{prefixMod g^{1}}{{Mod}_{g}}

…”

Section: A Birman Exact Sequence For Quotient Subgroupsmentioning

confidence: 99%

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Quotients of the mapping class group by power subgroups

Aramayona

Funar

2019

Bull. London Math. Soc.

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We study the quotient of the mapping class group Modgn of a surface of genus g with n punctures, by the subgroup prefixModgnfalse[pfalse] generated by the pth powers of Dehn twists. Our first main result is that prefixModg1/prefixModg1false[pfalse] contains an infinite normal subgroup of infinite index, and in particular is not commensurable to a higher rank lattice, for all but finitely many explicit values of p. Next, we prove that prefixModg0/prefixModg0false[pfalse] contains a Kähler subgroup of finite index, for every p⩾2 coprime with six. Finally, we observe that the existence of finite‐index subgroups of Modg0 with infinite abelianization is equivalent to the analogous problem for prefixModg0/prefixModg0false[pfalse].

show abstract

“…Finally, in order to prove (3), observe that

ϕ false(π_{1} (S_{g^{'}}) false) = π_{1} false(S_{g^{'}} false)

, by construction. In particular,

ϕ (π_{1} false(S_{g} false))

contains

π_{1} false(S_{g^{'}} false)

as a subgroup of finite index and we can again apply the result of Koberda–Santharoubane [, Theorem 4.1] used above to deduce that (3) holds. At this point, Lemma tells us that the group

Q_{1} : = false⟨ ⟨ {prefixproj}_{g}^{p} false(h^{p} false) ⟩ false⟩

is an infinite normal subgroup of infinite index of

{prefixMod}_{g}^{1} / {prefixMod}_{g}^{1} false[p false]

.…”

Section: Normal Subgroups Via Coversmentioning

confidence: 74%

“…We first explain how to construct the group

Q_{1}

from the statement. First, there exists

h_{1} \in π_{1} false(S_{g} false)

of infinite order in

{prefixMod}_{g}^{1} / {prefixMod}_{g}^{1} false[p false]

: this is a consequence of work by Koberda–Santharoubane [, Theorem 4.1] for large

p

, and Funar–Lochak [, Proof of Proposition 3.2] for all

p

as in the hypotheses of Theorem . We are going to produce an injective homomorphism

{prefixMod}_{g}^{1} \to {prefixMod}_{g^{'}}^{1}

as in Lemma , suited to this element

h_{1}

.…”

Section: Normal Subgroups Via Coversmentioning

confidence: 99%

“…Then the subgroups

{\bar{Q}}_{i} = Q_{i} \cap \frac{π_{1} false(S_{g} false)}{π_{1} false(S_{g} false) false[p false]}

are normal subgroups of

\frac{π_{1} (S_{g})}{π_{1} (S_{g}) [p]}

thus forming a descending normal series. Each

{\bar{Q}}_{i}

is infinite and of infinite index into

{\bar{Q}}_{i - 1}

by the proof above, based on the Koberda–Santharoubane infiniteness statement [, Theorem 1.1] . We have:

false⟨ ⟨ {prefixproj}_{g}^{p} false(h_{1}^{p} false), \dots, {prefixproj}_{g}^{p} false(h_{j}^{p} false) ⟩ false⟩ \subset \frac{π_{1} false(S_{g} false)}{π_{1} false(S_{g} false) false[p false]} .

In fact, the groups

⟨ false⟨ {proj}_{g}^{p} (h_{1}^{p}), \dots, {proj}_{g}^{p} (h_{j}^{p}) false⟩ ⟩

are characteristic subgroups of

\frac{π_{1} (S_{g})}{π_{1} (S_{g}) [p]}

, and in particular they are also normal subgroups in

\frac{prefixMod g^{1}}{{Mod}_{g}}

…”

Section: A Birman Exact Sequence For Quotient Subgroupsmentioning

confidence: 99%

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Quotients of the mapping class group by power subgroups

Aramayona

Funar

2019

Bull. London Math. Soc.

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“…However a normal discrete subgroup of SU g,p,ζ,(i) must be contained in its center, which is cyclic of order dim W g,p,(i) . Now, the result of [18] for i = 2 shows that the image of π g by ρ p,ζ,(i) is infinite non-abelian. We claim that this holds true for all i = 0 and we will give a detailed proof for i = p − 3.…”

Section: Quantum Surface Group Representationsmentioning

confidence: 97%

Profinite Completions of Burnside-Type Quotients of Surface Groups

Funar

Lochak

2018

Commun. Math. Phys.

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“…We now prove Theorems F and G via an argument of Koberda-Santharoubane [19]. We will give the details for Theorem G; the proof of Theorem F is similar.…”

Section: Integral Representations: the Proofs Of Theorems F And Gmentioning

confidence: 99%

Simple closed curves, finite covers of surfaces, and power subgroups of Out(Fn)

Malestein¹,

Putman²

2019

Duke Math. J.

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We construct examples of finite covers of punctured surfaces where the first rational homology is not spanned by lifts of simple closed curves. More generally, for any set O ⊂ F n which is contained in the union of finitely many Aut(F n )-orbits, we construct finiteindex normal subgroups of F n whose first rational homology is not spanned by powers of elements of O. These examples answer questions of Farb-Hensel, Kent, Looijenga, and Marché. We also show that the quotient of Out(F n ) by the subgroup generated by k th powers of transvections often contains infinite order elements, strengthening a result of Bridson-Vogtmann saying that it is often infinite. Finally, for any set O ⊂ F n which is contained in the union of finitely many Aut(F n )-orbits, we construct integral linear representations of free groups that have infinite image and map all elements of O to torsion elements.

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Quotients of surface groups and homology of finite covers via quantum representations

Cited by 34 publications

References 23 publications

Quotients of the mapping class group by power subgroups

Quotients of the mapping class group by power subgroups

Profinite Completions of Burnside-Type Quotients of Surface Groups

Simple closed curves, finite covers of surfaces, and power subgroups of Out(Fn)

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