On  right-angled Artin groups without surface subgroups

Kim, Sang-Hyun

doi:10.4171/ggd/84

Cited by 10 publications

(3 citation statements)

References 27 publications

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“…Many right-angled Artin groups contain quasi-isometrically embedded surface subgroups; see [40,11] (though the question of exactly which right-angled Artin groups contain surface subgroups is still open; see for example [16,23,24,10,39]). There are also constructions of surface subgroups of the mapping class group [1,26,15].…”

Section: Remarksmentioning

confidence: 99%

The geometry of right-angled Artin subgroups of mapping class groups

2012

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We describe sufficient conditions which guarantee that a finite set of mapping classes generate a right-angled Artin group quasi-isometrically embedded in the mapping class group. Moreover, under these conditions, the orbit map to Teichmüller space is a quasiisometric embedding for both of the standard metrics. As a consequence, we produce infinitely many genus h surfaces (for any h at least 2) in the moduli space of genus g surfaces (for any g at least 3) for which the universal covers are quasi-isometrically embedded in the Teichmüller space.

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Section: Remarksmentioning

confidence: 99%

The geometry of right-angled Artin subgroups of mapping class groups

2012

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“…Further sufficient conditions were found by Gordon and the author [14] and Kim and the author [23]. Kim and Oum answered Question 1 when is the double of a rank-two free group [22].…”

Section: Question 1 Does Every One-ended Hyperbolic Group Have a Surfmentioning

confidence: 88%

One-ended subgroups of graphs of free groups with cyclic edge groups

Wilton

2012

Geom. Topol.

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Consider a one-ended word-hyperbolic group. If it is the fundamental group of a graph of free groups with cyclic edge groups then either it is the fundamental group of a surface or it contains a finitely generated one-ended subgroup of infinite index. As a corollary, the same holds for limit groups. We also obtain a characterisation of surfaces with boundary among free groups equipped with peripheral structures.

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“…Sometimes called Baumslag's Lemma, this result plays a fundamental role in the theory of limit groups; see [105] and references therein. Baumslag's Lemma generalizes for multiple "twisting words" [69,5], and also for torsion-free wordhyperbolic groups [55]. We will present a continuous version of Baumslag's lemma in this section that will imply both of these generalizations.…”

Section: Lemma 31 ([6]mentioning

confidence: 97%

Flexibility of Group Actions on the Circle

Mj¹,

Koberda²,

Kim³

2019

Lecture Notes in Mathematics

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In this partly expository monograph we develop a general framework for producing uncountable families of exotic actions of certain classically studied groups acting on the circle. We show that if L is a nontrivial limit group then the nonlinear representation variety HompL, Homeo`pS 1 qq contains uncountably many semi-conjugacy classes of faithful actions on S 1 with pairwise disjoint rotation spectra (except for 0) such that each representation lifts to R. For the case of most Fuchsian groups L, we prove further that this flexibility phenomenon occurs even locally, thus complementing a result of K. Mann. We prove that each non-elementary free or surface group admits an action on S 1 that is never semi-conjugate to any action that factors through a finite-dimensional connected Lie subgroup in Homeo`pS 1 q. It is exhibited that the mapping class groups of bounded surfaces have non-semi-conjugate faithful actions on S 1 . In the process of establishing these results, we prove general combination theorems for indiscrete subgroups of PSL 2 pRq which apply to most Fuchsian groups and to all limit groups. We also show a Topological Baumslag Lemma, and general combination theorems for representations into Baire topological groups. The abundance of Z-valued subadditive defect-one quasimorphisms on these groups would follow as a corollary. We also give a mostly self-contained reconciliation of the various notions of semi-conjugacy in the extant literature by showing that they are all equivalent.

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On right-angled Artin groups without surface subgroups

Cited by 10 publications

References 27 publications

The geometry of right-angled Artin subgroups of mapping class groups

The geometry of right-angled Artin subgroups of mapping class groups

One-ended subgroups of graphs of free groups with cyclic edge groups

Flexibility of Group Actions on the Circle

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