Mapping class groups, multiple Kodaira fibrations, and CAT(0) spaces

Isenrich, Claudio Llosa; Py, Pierre

doi:10.1007/s00208-020-02125-y

Cited by 8 publications

(20 citation statements)

References 48 publications

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“…As [LIP21] pointed out, by [Mon06], in this case there exists a closed R × Λinvariant convex subset M ⊂ E which is minimal for these properties and canonical, and which is G-invariant. Moreover, the action of G on M is properly discontinuous and cocompact.…”

Section: Proof Of Theorem 15mentioning

confidence: 95%

“…Proof. The proof from [LIP21] that G Λ and G R act cocompactly and G Λ acts properly discontinuously works in this more general setting as written. Indeed, as the proof from [LIP21] points out that by [DK18, Theorem 5.67], the action of G Λ…”

Section: Proof Of Theorem 15mentioning

confidence: 98%

“…It follows from the proof of [LIP21, Theorem 3] with only minor changes. The proofs from [LIP21] of Lemma 3.1, Proposition 3.3 and Proposition 3.5 below work in our setting exactly as written. We omit some details which are included in [LIP21], the reader may find them in Section 3 of [LIP21].…”

Section: Proof Of Theorem 15mentioning

confidence: 99%

“…so that R acts isometrically on M 1 , Λ acts isometrically on M 2 and moreover the action of R × Λ on M is the product of these two actions. For details, see p.465 of [LIP21].…”

Section: Proof Of Theorem 15mentioning

confidence: 99%

“…(cf. [LIP21], Proposition 32): The G Λ-action on M 1 is properly discontinuous and cocompact. Similarly the G R-action on M 2 is properly discontinuous and cocompact.…”

Section: Proof Of Theorem 15mentioning

confidence: 99%

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Conditions on the monodromy for a surface group extension to be CAT(0)

Zhu¹

2022

Preprint

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In order to determine when surface-by-surface bundles are non-positively curved, Llosa Isenrich and Py in [LIP21] give a necessary condition: given a surfaceby-surface group G with infinite monodromy, if G is CAT(0) then the monodromy representation is injective. We extend this to a more general result: Let G be a group with a normal surface subgroup R. Assume G R satisfies the property that for every infinite normal subgroup Λ of G R, there is an infinite subgroup Λ 0 < Λ so that the centralizer C G R (Λ 0 ) is finite. If G is CAT(0) with infinite monodromy, then the monodromy representation has finite kernel. We prove that acylindrically hyperbolic groups satisfy this property.

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Section: Proof Of Theorem 15mentioning

confidence: 95%

Section: Proof Of Theorem 15mentioning

confidence: 98%

Section: Proof Of Theorem 15mentioning

confidence: 99%

“…so that R acts isometrically on M 1 , Λ acts isometrically on M 2 and moreover the action of R × Λ on M is the product of these two actions. For details, see p.465 of [LIP21].…”

Section: Proof Of Theorem 15mentioning

confidence: 99%

“…(cf. [LIP21], Proposition 32): The G Λ-action on M 1 is properly discontinuous and cocompact. Similarly the G R-action on M 2 is properly discontinuous and cocompact.…”

Section: Proof Of Theorem 15mentioning

confidence: 99%

See 3 more Smart Citations

Conditions on the monodromy for a surface group extension to be CAT(0)

Zhu¹

2022

Preprint

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Conditions on the monodromy for a surface group extension to be CAT(0)

Zhu

2023

Proc. Amer. Math. Soc.

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In order to determine when surface-by-surface bundles are non-positively curved, Llosa Isenrich and Py [Math. Ann. 380 (2021), pp. 449–485] give a necessary condition: given a surface-by-surface group G G with infinite monodromy, if G G is CAT(0) then the monodromy representation is injective. We extend this to a more general result: Let G G be a group with a normal surface subgroup R R . Assume G / R G/R satisfies the property that for every infinite normal subgroup Λ \Lambda of G / R G/R , there is an infinite finitely generated subgroup Λ 0 > Λ \Lambda _0>\Lambda so that the centralizer C G / R ( Λ 0 ) C_{G/R}(\Lambda _0) is finite. We then prove that if G G is CAT(0) with infinite monodromy, then the monodromy representation has a finite kernel. This applies in particular if G / R G/R is acylindrically hyperbolic.

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