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PreprintarXiv licencearXiv

Infranormal Kazhdan subgroups and instability in finite actions

  • Jonas Pinke
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Abstract

We prove that the group G=ℤ[1/2]3rtimes(SL3(ℤ)×ℤ) is finitely presented and residually finite, but is not stable in finite actions, answering a question of Gohla and Thom. The proof uses the infranormal Kazhdan subgroup H=ℤ3rtimesSL3(ℤ) and its normal closure N in G. A Kazhdan projection estimate shows that, in any G-action weakly contained in finite actions, every H-invariant measurable subset is also N-invariant. We construct an explicit sofic approximation of G whose limit action contains a measurable subset that is H-invariant but not N-invariant.

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Cite

BibTeXRIS
BibTeX
@misc{pinke2026infranormal,
  title = {{Infranormal Kazhdan subgroups and instability in finite actions}},
  author = {Jonas Pinke},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.35841},
  eprint = {2610.35841},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.35841},
}
RIS
TY  - UNPB
TI  - Infranormal Kazhdan subgroups and instability in finite actions
AU  - Jonas Pinke
PY  - 2026
DA  - 2026/10/26
PB  - arXiv
DO  - 10.48550/arxiv.2610.35841
UR  - https://arxiv.org/abs/2610.35841
AB  - We prove that the group G=ℤ[1/2]^3rtimes(SL_3(ℤ)×ℤ) is finitely presented and residually finite, but is not stable in finite actions, answering a question of Gohla and Thom. The proof uses the infranormal Kazhdan subgroup H=ℤ^3rtimesSL_3(ℤ) and its normal closure N in G. A Kazhdan projection estimate shows that, in any G-action weakly contained in finite actions, every H-invariant measurable subset is also N-invariant. We construct an explicit sofic approximation of G whose limit action contains a measurable subset that is H-invariant but not N-invariant.
ER  -