Residually finite rationally p groups

Koberda, Thomas; Suciu, Alexander I.

doi:10.1142/s0219199719500160

Cited by 5 publications

(8 citation statements)

References 43 publications

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“…Both of the conclusions of Proposition 3.12 hold for general right-angled Artin groups, though we will not need such generality here. See [11,127,149] for a discussion of two-generated subgroups, and [62,145] for a discussion of Hopficity and residual finiteness. The proof of Proposition 3.12 draws on ideas in [62,145] especially.…”

Section: Claim: the Collectionmentioning

confidence: 99%

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Structure and regularity of group actions on one-manifolds

Kim¹,

Koberda²

2021

Preprint

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This book represents an account and contextualization of a research program that was executed by the authors and their collaborators over the period of several years. Both authors began their careers in classical geometric group theory and began collaborating on projects about right-angled Artin groups and their relationship with mapping class groups of surfaces, at a time when the first author was a visiting assistant professor at Tufts University and while the second author was a graduate student at Harvard University.Around 2014, the authors became interested in a question attributed by M. Kapovich to V. Kharlamov, one that asks which right-angled Artin groups can act faithfully by smooth diffeomorphisms on the circle. Because of the profusion of right-angled Artin subgroups of mapping class groups and because of the robust persistence of right-angled Artin groups under passing to finite index subgroups, an answer to Kharlamov's question had the potential to shed light on a question of F. Labourie that he posed in his 1998 ICM talk and that was reiterated by B. Farb and J. Franks in several places: are there finite index subgroups of (sufficiently complicated) mapping class groups acting smoothly on the circle?At first, it seemed to the authors that an easy solution to Kharlamov's question was available, and that all right-angled Artin groups admitted such faithful actions. It was soon pointed out by J. Bowden that the actions constructed by the authors and H. Baik in fact failed to be differentiable at certain accumulations of fixed points, and so only furnished faithful smooth actions on the real line. After approximately two years of work, inspired by Bowden's observation, Baik and the authors managed to prove that that most right-angled Artin groups do not admit faithful C 2 actions on the compact interval nor on the circle. As a consequence, Labourie's question could be completely answered in the case of C 2 actions of finite index subgroups of mapping class groups.About a year and a half later, the authors were able to develop a tool that they dubbed the abt-Lemma, which is a certain combinatorial-algebraic obstruction for smoothness of a group action on a compact one-manifold. As a result, they were able to answer Kharlamov's question by giving a concise characterization of rightangled Artin groups acting faithfully by smooth diffeomorphisms on the circle.Because of the technical details involved in the analysis of smooth actions of right-angled Artin groups on one-manifolds, the authors became interested in Thompson's group F , which occurs naturally in these contexts. Inspired by the dynamical theory of Thompson's group, the authors investigated the class of chain groups in subsequent work with Y. Lodha. These objects form a highly diverse class of groups of homeomorphisms of the interval, while nevertheless exhibiting remarkable uniformity properties. i v Contents Preface i Acknowledgements v Chapter 1. Introduction Appendix B. Orderability and Hölder's Theorem 1. Orderability of groups and homeomorp...

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Section: Claim: the Collectionmentioning

confidence: 99%

“…See [11,127,149] for a discussion of two-generated subgroups, and [62,145] for a discussion of Hopficity and residual finiteness. The proof of Proposition 3.12 draws on ideas in [62,145] especially.…”

Section: Claim: the Collectionmentioning

confidence: 99%

Structure and regularity of group actions on one-manifolds

Kim¹,

Koberda²

2021

Preprint

Self Cite

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“…Now let

π

be a finitely generated group, and define

V_{r}^{i} false(π false) : = V_{r}^{i} false(K (π, 1) false)

for

i, r ⩾ 0

. It is known that the sets

V_{r}^{i} false(π false)

with

i ⩽ 1

and

r ⩾ 0

depend only on the maximal metabelian quotient

π / π^{''}

(see for example, ); more precisely, the following equality holds,

V_{r}^{i} false(π false) = V_{r}^{i} false(π / π^{''} false) .

…”

Section: Cohomology Jump Loci Finiteness Properties and Largenessmentioning

confidence: 99%

“…Also in , the notion of RFR

p

group (for

p

a prime) is introduced. The class of groups which are RFR

p

for all primes

p

includes all groups of the form

π_{1} false(C false)

, for

C

a smooth complex curve with

χ (C) < 0

, and all right‐angled Artin groups.…”

Section: Cohomology Jump Loci Finiteness Properties and Largenessmentioning

confidence: 99%

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Infinitesimal finiteness obstructions

Papadima

Suciu

2018

J. London Math. Soc.

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Does a space enjoying good finiteness properties admit an algebraic model with commensurable finiteness properties? In this note, we provide a rational homotopy obstruction for this to happen. As an application, we show that the maximal metabelian quotient of a very large, finitely generated group is not finitely presented. Using the theory of 1‐minimal models, we also show that a finitely generated group π admits a connected 1‐model with finite‐dimensional degree 1 piece if and only if the Malcev Lie algebra m(π) is the lower central series completion of a finitely presented Lie algebra.

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Homological growth of Artin kernels in positive characteristic

Hughes

Leary

2023

Math. Ann.

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We prove an analogue of the Lück Approximation Theorem in positive characteristic for certain residually finite rationally soluble (RFRS) groups including right-angled Artin groups and Bestvina–Brady groups. Specifically, we prove that the mod p homology growth equals the dimension of the group homology with coefficients in a certain universal division ring and this is independent of the choice of residual chain. For general RFRS groups we obtain an inequality between the invariants. We also consider a number of applications to fibring, amenable category, and minimal volume entropy.

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Residually finite rationally p groups

Cited by 5 publications

References 43 publications

Structure and regularity of group actions on one-manifolds

Structure and regularity of group actions on one-manifolds

Infinitesimal finiteness obstructions

Homological growth of Artin kernels in positive characteristic

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