Property (T) and nonlinearity of mapping class group quotients

Abstract

We give a general method for proving Kazhdan's property (T) for quotients G/K_[c+1], where G is countable, K is a normal subgroup and K_[j] denotes its lower central series. The method combines an affine realization of G/[K,K], a contraction argument, and permanence for nilpotent normal subgroups. We apply this method to several classes of groups by means of homological representations. For the Torelli lower-central series quotients Mod(Σ_g)/(𝒯_g)_[c+1] for g≥3 we show that they have property (T) for every c≥1. We also prove property (T) for Aut(F₃)/(IA₃)_[c+1] for every c≥1, relating these groups to the tame nilpotent images studied by Lubotzky and Pak. Several other classes of examples are discussed as well. We also prove that the Torelli lower-central series quotients for c≥2 are not residually finite, and thus non-linear.

Groups with property (T) and many alternating group quotients
Laurent Bartholdi, Martin Kassabov
Groups, Geometry, and Dynamics3 Nov 2026Peer-reviewedCC BYFull textAlgebra & number theoryHAL
Property P_naive for non-orientable big Mapping Class Groups
Jesús Hernández Hernández, Tianyi Lou
arXiv5 Sep 2026PreprintarXiv licenceFull textAlgebra & number theory Geometry & topologyarXiv
A Faithful Action of the Mapping Class Group Mod(S_g) on the (2g−1)-Sphere
Shile Wang
arXiv5 Oct 2026PreprintarXiv licenceFull textGeometry & topologyarXiv
Generating the twist subgroup of the level $2$ mapping class group of a non-orientable closed surface
Ryoma Kobayashi
arXiv16 Sep 2026PreprintarXiv licenceFull textAlgebra & number theory Geometry & topologyarXiv
An explicit section of the Laudenbach-type exact sequence of the big mapping class group of M_Γ
Jorge Andres Robinson Arrieta
arXiv17 Oct 2026PreprintarXiv licenceFull textAlgebra & number theoryarXiv

Cite

BibTeX RIS

BibTeX

@misc{nowak2026property,
  title = {{Property $(T)$ and nonlinearity of mapping class group quotients}},
  author = {Piotr W. Nowak},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2609.12196},
  eprint = {2609.12196},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.12196},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Property (T) and nonlinearity of mapping class group quotients
AU  - Piotr W. Nowak
PY  - 2026
DA  - 2026/10/14
PB  - arXiv
DO  - 10.48550/arxiv.2609.12196
UR  - https://arxiv.org/abs/2609.12196
AB  - We give a general method for proving Kazhdan's property (T) for quotients G/K_[c+1], where G is countable, K is a normal subgroup and K_[j] denotes its lower central series. The method combines an affine realization of G/[K,K], a contraction argument, and permanence for nilpotent normal subgroups. We apply this method to several classes of groups by means of homological representations. For the Torelli lower-central series quotients Mod(Σ_g)/(𝒯_g)_[c+1] for g≥3 we show that they have property (T) for every c≥1. We also prove property (T) for Aut(F_3)/(IA_3)_[c+1] for every c≥1, relating these groups to the tame nilpotent images studied by Lubotzky and Pak. Several other classes of examples are discussed as well. We also prove that the Torelli lower-central series quotients for c≥2 are not residually finite, and thus non-linear.
ER  -

Abstract

Related

Cite