Shalom’s property HFD and extensions by ℤ of locally finite groups

Brieussel, Jérémie; Zheng, Tianyi

doi:10.1007/s11856-018-1818-6

Cited by 5 publications

(7 citation statements)

References 18 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For instance, the standard ( restricted ) wreath product

(Z / 2 Z) ≀ Z

, see Subsection 4.5 for the definition, is a locally‐finite‐lift of

double-struckZ

(it is, moreover, a local‐normal‐finite‐lift of

double-struckZ

in the sense of ). However, it is not an LFNF‐lift of

double-struckZ

.…”

Section: Application To ‘Amenability Versus Non‐exactness'mentioning

confidence: 99%

“…Note that local‐full‐normal‐finiteness implies local finiteness. Unlike local finiteness (or local‐normal‐finiteness in the sense of ), this property is not ‘intrinsic’, namely, this is a property as a subgroup

N

normalΛ

, but of a group

N

alone.…”

Section: Application To ‘Amenability Versus Non‐exactness'mentioning

confidence: 99%

“…As a by‐product of it, we show that the group

Λ_{1}

appearing in Theorem may be arranged such that it has the Liouville property ( with respect to every symmetric finite generating set ) and Shalom's property

H_{FD}

. We do not recall the definitions of these properties; see for the definitions and more details. We make a remark that among amenable groups, groups with these properties are, respectively, considered as ‘small’ groups in certain senses.…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…We make a remark that among amenable groups, groups with these properties are, respectively, considered as ‘small’ groups in certain senses. For instance, it is known by [, Proposition 6.4 ] that

false(double-struckZ / 2 double-struckZ false) ≀ {double-struckZ}^{3}

does not have the Liouville property; in [, Proposition 4.4 ], it is showed that

{Sym}_{< ℵ_{0}} false(double-struckZ false) ⋊ Z

fails to have property

H_{FD}

.…”

Section: Proof Of Theoremmentioning

confidence: 99%

See 3 more Smart Citations

Amenability versus non‐exactness of dense subgroups of a compact group

Mimura

2019

J. London Math. Soc.

View full text Add to dashboard Cite

Given a countable residually finite group, we construct a compact group K and two elements w and u of K with the following properties: The group generated by w and u3 is amenable, the group generated by w and u contains a copy of the given group, and these two groups are dense in K. By combining it with a construction of non‐exact groups that are locally embeddable into finite (LEF) by Osajda and Arzhantseva–Osajda and formation of diagonal products, we construct an example for which the latter dense group is non‐exact. Our proof employs approximations in the space of marked groups of LEF groups.

show abstract

“…For instance, the standard ( restricted ) wreath product

(Z / 2 Z) ≀ Z

, see Subsection 4.5 for the definition, is a locally‐finite‐lift of

double-struckZ

(it is, moreover, a local‐normal‐finite‐lift of

double-struckZ

in the sense of ). However, it is not an LFNF‐lift of

double-struckZ

.…”

Section: Application To ‘Amenability Versus Non‐exactness'mentioning

confidence: 99%

N

normalΛ

, but of a group

N

alone.…”

Section: Application To ‘Amenability Versus Non‐exactness'mentioning

confidence: 99%

“…As a by‐product of it, we show that the group

Λ_{1}

appearing in Theorem may be arranged such that it has the Liouville property ( with respect to every symmetric finite generating set ) and Shalom's property

H_{FD}

Section: Proof Of Theoremmentioning

confidence: 99%

false(double-struckZ / 2 double-struckZ false) ≀ {double-struckZ}^{3}

does not have the Liouville property; in [, Proposition 4.4 ], it is showed that

{Sym}_{< ℵ_{0}} false(double-struckZ false) ⋊ Z

fails to have property

H_{FD}

.…”

Section: Proof Of Theoremmentioning

confidence: 99%

See 2 more Smart Citations

Amenability versus non‐exactness of dense subgroups of a compact group

Mimura

2019

J. London Math. Soc.

View full text Add to dashboard Cite

show abstract

“…If π is weakly mixing and there is an infinite H⊳G with G/H cyclic and H 1 (H, π |H ) = 0, does H 1 (G, π) = 0? Question 6.6.4 has been answered in the negative by Brieussel & Zheng [8,Remark 4.6]. In Question 6.6.1-3, it would be reasonable to add hypothesis such as G = H, g 0 or H 1 (G, π) = H 1 (G, π) or strongly mixing.…”

mentioning

confidence: 99%

Mixing, malnormal subgroups and cohomology in degree one

Gournay

2018

Groups Geom. Dyn.

View full text Add to dashboard Cite

The aim of the current paper is to explore the implications on the group G of the non-vanishing of the cohomology in degree one of one of its representation π, given some mixing conditions on π. For example, harmonic cocycles of weakly mixing unitary representations factorise by the FC-centre. In that case non-vanishing implies the FC-centre is trivial or fixes a vector. Next, for any subgroup H < G, H will either be "small", almost-malnormal or π |H also has non-trivial cohomology in degree one (in this statement, "small", reduced vs unreduced cohomology and unitary vs generic depend on the mixing condition). The notion of q-normal subgroups is an important ingredient of the proof and results on the vanishing of the reduced ℓ p -cohomology in degree one are obtained as an intermediate step.

show abstract

The Poisson boundary of lampshuffler groups

Silva

2024

Math. Z.

View full text Add to dashboard Cite

We study random walks on the lampshuffler group $$\textrm{FSym}(H)\rtimes H$$ FSym ( H ) ⋊ H , where H is a finitely generated group and $$\textrm{FSym}(H)$$ FSym ( H ) is the group of finitary permutations of H. We show that for any step distribution $$\mu $$ μ with a finite first moment that induces a transient random walk on H, the permutation coordinate of the random walk almost surely stabilizes pointwise. Our main result states that for $$H=\mathbb {Z}$$ H = Z , the above convergence completely describes the Poisson boundary of the random walk $$(\textrm{FSym}(\mathbb {Z})\rtimes \mathbb {Z},\mu )$$ ( FSym ( Z ) ⋊ Z , μ ) .

show abstract

Shalom’s property HFD and extensions by ℤ of locally finite groups

Cited by 5 publications

References 18 publications

Amenability versus non‐exactness of dense subgroups of a compact group

Amenability versus non‐exactness of dense subgroups of a compact group

Mixing, malnormal subgroups and cohomology in degree one

The Poisson boundary of lampshuffler groups

Contact Info

Product

Resources

About