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A finitely generated ICC non property (T) group that is neither inner amenable nor properly proximal
Abstract
This note record a finitely generated extension of Caprace's example of a non-inner-amenable group is also not properly proximal, which answers the original question by Boutonnet, Ioana and Peterson. The extension is an explicit ICC subgroup of GL4(𝔽2(t)), which has no property \textup{(T)}. Failure of proper proximality follows from the work of Ishan-Peterson-Ruth and the passage of proper proximality to co-amenable subgroups. Non-inner-amenability follows from a property-\textup{(T)} subgroup and an elementary centralizer computation.
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BibTeX
@misc{ma2026finitely,
title = {{A finitely generated ICC non property (T) group that is neither inner amenable nor properly proximal}},
author = {Xin Ma},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.30158},
eprint = {2609.30158},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.30158},
}
RIS
TY - UNPB
TI - A finitely generated ICC non property (T) group that is neither inner amenable nor properly proximal
AU - Xin Ma
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.30158
UR - https://arxiv.org/abs/2609.30158
AB - This note record a finitely generated extension of Caprace's example of a non-inner-amenable group is also not properly proximal, which answers the original question by Boutonnet, Ioana and Peterson. The extension is an explicit ICC subgroup of GL_4(𝔽_2(t)), which has no property \textup{(T)}. Failure of proper proximality follows from the work of Ishan-Peterson-Ruth and the passage of proper proximality to co-amenable subgroups. Non-inner-amenability follows from a property-\textup{(T)} subgroup and an elementary centralizer computation.
ER -