Lamplighters and the bounded cohomology of Thompson’s group

Monod, Nicolas

doi:10.1007/s00039-022-00604-9

Cited by 8 publications

(2 citation statements)

References 25 publications

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“…While in this paper, we focus on using the theory of asymptotic cohomology to prove uniform Ustability for lattices in semisimple groups, the framework and tools developed here have also been used in subsequent work [FFR23] to prove uniform U-stability for other classes of groups such as lamplighter groups Γ ≀ Λ where Λ is infinite and amenable, as well as several groups of dynamical origin such as Thompson's group F . The techniques there too are analogous to corresponding vanishing results of bounded cohomology in [Mon22], yet again highlighting the connections between the theories of bounded cohomology and asymptotic cohomology.…”

Section: Main Results and Methodsmentioning

confidence: 77%

Asymptotic Cohomology and Uniform Stability for Lattices in Semisimple Groups

Glebsky¹,

Lubotzky²,

Monod³

et al. 2023

Preprint

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It is, by now, classical that lattices in higher rank semisimple groups have various rigidity properties. In this work, we add another such rigidity property to the list, namely uniform stability with respect to the family of unitary operators on finite-dimensional Hilbert spaces equipped with submultiplicative norms. Towards this goal, we first build an elaborate cohomological theory capturing the obstruction to such stability, and show that the vanishing of second cohomology implies uniform stability in this setting. This cohomology can be roughly thought of as an asymptotic version of bounded cohomology, and sheds light on a question raised in [Mon06] about a possible connection between vanishing of second bounded cohomology and Ulam stability. Along the way, we use this criterion to provide a short conceptual (re)proof of the classical result of Kazhdan [Kaz82] that discrete amenable groups are Ulam stable. We then use this machinery to establish our main result, that lattices in a class of higher rank semisimple groups (which are known to have vanishing bounded cohomology) are uniformly stable.

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Section: Main Results and Methodsmentioning

confidence: 77%

Asymptotic Cohomology and Uniform Stability for Lattices in Semisimple Groups

Glebsky¹,

Lubotzky²,

Monod³

et al. 2023

Preprint

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show abstract

“…The characterisation in Theorem 1.7 allows us to establish that many groups admit a positive answer to Question 1. • all amenable groups of type FH d because they have trivial bounded cohomology [9,14] (and thus SV (d) = {0} [9]); • more generally, all boundedly acyclic groups of type FH d ; this includes the Thompson group F [22]; • all hyperbolic groups because they are of finite type and the 1 -semi-norm is a norm by the duality principle and Mineyev's results [19];…”

Section: Examplesmentioning

confidence: 99%

The spectrum of simplicial volume with fixed fundamental group

Löh

2022

Geom Dedicata

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Ulam stability of lamplighters and Thompson groups

Fournier-Facio,

Rangarajan

2023

Math. Ann.

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We show that a large family of groups is uniformly stable relative to unitary groups equipped with submultiplicative norms, such as the operator, Frobenius, and Schatten p-norms. These include lamplighters $$\Gamma \wr \Lambda $$ Γ ≀ Λ where $$\Lambda $$ Λ is infinite and amenable, as well as several groups of dynamical origin such as the classical Thompson groups $$F, F', T$$ F , F ′ , T and V. We prove this by means of vanishing results in asymptotic cohomology, a theory introduced by the second author, Glebsky, Lubotzky and Monod, which is suitable for studying uniform stability. Along the way, we prove some foundational results in asymptotic cohomology, and use them to prove some hereditary features of Ulam stability. We further discuss metric approximation properties of such groups, taking values in unitary or symmetric groups.

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Lamplighters and the bounded cohomology of Thompson’s group

Cited by 8 publications

References 25 publications

Asymptotic Cohomology and Uniform Stability for Lattices in Semisimple Groups

Asymptotic Cohomology and Uniform Stability for Lattices in Semisimple Groups

The spectrum of simplicial volume with fixed fundamental group

Ulam stability of lamplighters and Thompson groups

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