Finitely generated groups acting uniformly properly on hyperbolic space

Kropholler, Robert; Vankov, Vladimir

doi:10.48550/arxiv.2007.13880

“…Some cases of the first part of Theorem 1.2 appeared as [14, thm. 3.1], and conjecture 1.3 of [9] discusses another context in which groups that are parametrized by subsets of Z are expected to be virtually torsion-free if and only if the subset is periodic; interestingly the opposite implication is the one that remains open for those groups. In the 1970's Dyson defined a family of groups L(S) for S ⊆ Z as amalgamations of two copies of the lamplighter group, and she showed that L(S) is residually finite if and only if S is closed in Z [5].…”

Section: Conjecture 14 If S Is Closed In the Profinite Topology On Z ...mentioning

confidence: 99%

On the virtual and residual properties of a generalization of Bestvina-Brady groups

Leary

¹

,

Vankov

²

2022

Self Cite

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Previously one of us introduced a family of groups $$G^M_L(S)$$ G L M ( S ) , parametrized by a finite flag complex L, a regular covering M of L, and a set S of integers. We give conjectural descriptions of when $$G^M_L(S)$$ G L M ( S ) is either residually finite or virtually torsion-free. In the case that M is a finite cover and S is periodic, there is an extension with kernel $$G_L^M(S)$$ G L M ( S ) and infinite cyclic quotient that is a CAT(0) cubical group. We conjecture that this group is virtually special. We relate these three conjectures to each other and prove many cases of them.

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“…Some cases of the first part of Theorem 1.2 appeared as [23, thm. 3.1], and conjecture 1.3 of [15] discusses another context in which groups that are parametrized by subsets of Z are expected to be virtually torsion-free if and only if the subset is periodic; interestingly the opposite implication is the one that remains open for those groups. In the 1970's Dyson defined a family of groups L(S) for S ⊆ Z as amalgamations of two copies of the lamplighter group, and she showed that L(S) is residually finite if and only if S is closed in Z [11].…”

Section: If G Mmentioning

confidence: 99%

On subamalgams of partially ordered monoids

Subaiei

¹

,

Renshaw²

2022

Semigroup Forum

1

0

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Previously one of us introduced a family of groups G M L (S), parametrized by a finite flag complex L, a regular covering M of L, and a set S of integers. We give conjectural descriptions of when G M L (S) is either residually finite or virtually torsion-free. In the case that M is a finite cover and S is periodic, there is an extension with kernel G M L (S) and infinite cyclic quotient that is a CAT(0) cubical group. We conjecture that this group is virtually special. We relate these three conjectures to each other and prove many cases of them.

show abstract

On the virtual and residual properties of a generalization of Bestvina-Brady groups

Leary¹,

Vankov²

2022

Preprint

Self Cite

0

View full text Add to dashboard Cite

Previously one of us introduced a family of groups G M L (S), parametrized by a finite flag complex L, a regular covering M of L, and a set S of integers. We give conjectural descriptions of when G M L (S) is either residually finite or virtually torsion-free. In the case that M is a finite cover and S is periodic, there is an extension with kernel G M L (S) and infinite cyclic quotient that is a CAT(0) cubical group. We conjecture that this group is virtually special. We relate these three conjectures to each other and prove many cases of them.

show abstract

Finitely generated groups acting uniformly properly on hyperbolic space

Cited by 3 publications

References 11 publications

On the virtual and residual properties of a generalization of Bestvina-Brady groups

On the virtual and residual properties of a generalization of Bestvina-Brady groups

On subamalgams of partially ordered monoids

On the virtual and residual properties of a generalization of Bestvina-Brady groups

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