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Two-generator Kazhdan-constant question for SL_n(Z)

For each integer n≥3n\geq 3, let G=SLn(Z)G=\mathrm{SL}_n(\mathbb{Z}) and, for a finite generating set S⊆GS\subseteq G, let κ(G,S)\kappa(G,S) denote its Kazhdan constant. Is inf⁡{κ(G,S):S⊆G, ∣S∣=2, ⟨S⟩=G}\inf\{\kappa(G,S): S\subseteq G,\ |S|=2,\ \langle S\rangle=G\} strictly positive? Equivalently, does there exist, for each n≥3n\geq 3, a constant cn>0c_n>0 such that every two-element generating set SS of SLn(Z)\mathrm{SL}_n(\mathbb{Z}) satisfies κ(SLn(Z),S)≥cn\kappa(\mathrm{SL}_n(\mathbb{Z}),S)\geq c_n?

References

Primary source

arXiv

Additional references

geometric-group-theoryrepresentation-theory
Posted · edited

Progress summary

Refreshed
Claimed solved

A new unrefereed preprint claims that even two generators cannot give a uniform spectral gap for these groups.

The question asks whether Kazhdan constants for generating sets of exactly two elements of SLn(Z)\mathrm{SL}_n(\mathbb{Z}) can have a positive uniform lower bound. A September 2026 preprint claims the infimum is zero.

Known results

Earlier work gives lower bounds for the standard elementary-matrix generating set of SLn(Z)\mathrm{SL}_n(\mathbb{Z}), including a bound of (42n+860)−1(42\sqrt{n}+860)^{-1}. Related work shows vanishing Kazhdan constants for suitable finite-index subgroups and fixed generating-set sizes, but does not establish the exact two-generator statement for SLn(Z)\mathrm{SL}_n(\mathbb{Z}).

September 2026 claimed resolution

Jvbin Yao's preprint Kazhdan constants for two-element generating sets of SLn(Z)\mathrm{SL}_n(\mathbb{Z}) claims that the infimum is zero even when the generating sets are required to have exactly two elements. This would settle the tracked question negatively, but the claim is unrefereed and unverified.

Current status (as of September 2026): The exact two-generator claim is asserted by a new preprint but remains unverified; no independently confirmed resolution was found.

Sources

Solutions 0

No solutions have been posted yet.