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Kazhdan constants for two-element generating sets of SLn(ℤ)

  • Jvbin Yao
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Abstract

For n≥3, the group SLn(ℤ) has property (T), so every finite generating set has a positive Kazhdan constant. However, it was recently shown that the infimum of these constants over all finite generating sets is $0$. This naturally raises the question of whether the infimum remains zero when the cardinality of the generating sets is bounded in advance. We prove that the infimum of the Kazhdan constants of SLn(ℤ) over all two-element generating sets is $0$ for every n≥3.

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Cite

BibTeXRIS
BibTeX
@misc{yao2026kazhdan,
  title = {{Kazhdan constants for two-element generating sets of $\mathrmSL_n(\mathbb Z)$}},
  author = {Jvbin Yao},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.16631},
  eprint = {2609.16631},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.16631},
}
RIS
TY  - UNPB
TI  - Kazhdan constants for two-element generating sets of SL_n(ℤ)
AU  - Jvbin Yao
PY  - 2026
DA  - 2026/09/15
PB  - arXiv
DO  - 10.48550/arxiv.2609.16631
UR  - https://arxiv.org/abs/2609.16631
AB  - For n≥3, the group SL_n(ℤ) has property (T), so every finite generating set has a positive Kazhdan constant. However, it was recently shown that the infimum of these constants over all finite generating sets is $0$. This naturally raises the question of whether the infimum remains zero when the cardinality of the generating sets is bounded in advance. We prove that the infimum of the Kazhdan constants of SL_n(ℤ) over all two-element generating sets is $0$ for every n≥3.
ER  -