Uniform simplicity of groups with proximal action

Gal, Światosław R.; Gismatullin, Jakub

doi:10.1090/btran/18

Cited by 17 publications

(17 citation statements)

References 26 publications

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“…It is easy to see that uniformly simple groups are uniformly perfect. The following is a special case of Theorem 1.1 in [28].…”

Section: A Group Actionmentioning

confidence: 95%

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Second bounded cohomology of groups acting on $1$-manifolds and applications to spectrum problems

Fournier-Facio¹,

Lodha²

2021

Preprint

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We prove a general criterion for the vanishing of second bounded cohomology (with trivial real coefficients) for groups that admit an action satisfying certain mild hypotheses. This leads to new computations of the second bounded cohomology for a large class of groups of homeomorphisms of 1-manifolds, and a plethora of applications. First, we demonstrate that the finitely presented and nonamenable group G 0 constructed by the second author with Justin Moore satisfies that every subgroup has vanishing second bounded cohomology. This provides the first solution to the so-called homological von Neumann-Day Problem, as discussed by Calegari. Then we provide the first examples of finitely presented groups whose spectrum of stable commutator length contains algebraic irrationals, answering a question of Calegari. Next, we provide the first examples of finitely generated left orderable groups that are not locally indicable, and yet have vanishing second bounded cohomology. This proves that a homological analogue of the Witte-Morris Theorem does not hold. Finally, we combine some of the aforementioned results to provide the first examples of manifolds whose simplicial volumes are algebraic and irrational. This is further evidence towards a conjecture of Heuer and Löh.

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“…It is easy to see that uniformly simple groups are uniformly perfect. The following is a special case of Theorem 1.1 in [28].…”

Section: A Group Actionmentioning

confidence: 95%

“…Theorem 2.1 (Gal, Gismatullin [28]). Let G ≤ Homeo + (R) be a boundedly supported and promixal group action.…”

Section: A Group Actionmentioning

confidence: 99%

Second bounded cohomology of groups acting on $1$-manifolds and applications to spectrum problems

Fournier-Facio¹,

Lodha²

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…It is easy to see that uniformly simple groups are uniformly perfect. The following is a special case of Theorem 1.1 in [13].…”

Section: Introductionmentioning

confidence: 95%

“…Theorem 2.1 (Gal, Gismatullin [13]). Let G ≤ Homeo + (R) be a boundedly supported and promixal group action.…”

Section: Introductionmentioning

confidence: 99%

Algebraic irrational stable commutator length in finitely presented groups

Fournier-Facio¹,

Lodha²

2021

Preprint

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We provide the first example of a finitely presented (in fact, type F ∞ ) group with elements whose stable commutator length is algebraic and irrational, answering a question of Calegari. Our example is the lift to the real line of the golden ratio Thompson group T τ : the circle analogue of the Cleary's golden ratio Thompson group F τ which acts on the interval. PreliminariesGroup actions will always be on the right. Accordingly, we will use the convention [g, h] = g −1 h −1 gh for commutators. Dynamics of group actions on 1-manifoldsGiven g ∈ Homeo + (M) for a given 1-manifold M, we define the support of g as:We identify S 1 with R/Z. Recall that Homeo + (R) and Homeo + (R/Z) are the groups of orientation preserving homeomorphisms of the real line and the circle. The latter group

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“…Note that for the groups that we are interested in here, any normal generation function is unbounded. Examples of groups with uniform bound on the normal generation function are given in [5,6,11,17].…”

Section: Introductionmentioning

confidence: 99%

Bounded Normal Generation for Projective Unitary Groups of Certain Infinite Operator Algebras

Dowerk

Thom

2017

International Mathematics Research Notices

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We study the question how quickly products of a fixed conjugacy class cover the entire group in the projective unitary group of the connected component of the identity of the Calkin algebra, as well as the projective unitary group of a factor von Neumann algebra of type III. Our result is that the number of factors that are needed is as small as permitted by the (essential) operator norm -in analogy to a result of Liebeck-Shalev for non-abelian finite simple groups and analogous results for unitary groups of II1-factors.

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Uniform simplicity of groups with proximal action

Cited by 17 publications

References 26 publications

Second bounded cohomology of groups acting on $1$-manifolds and applications to spectrum problems

Second bounded cohomology of groups acting on $1$-manifolds and applications to spectrum problems

Algebraic irrational stable commutator length in finitely presented groups

Bounded Normal Generation for Projective Unitary Groups of Certain Infinite Operator Algebras

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