Uniform filling conjecture for finite quotients of Bruhat–Tits buildings

About 13 years old · traced to

Let B\mathcal{B} be the Bruhat–Tits building associated with PGLd(F)\mathrm{PGL}_{d}(F), where FF is a local field and d≥3d\geq3. For a finite quotient XX of B\mathcal{B}, let νi(X)\nu_i(X) denote its filling in dimension ii. Uniform filling conjecture. There exists a constant ν=ν(d,F)\nu=\nu(d,F) such that

νi(X)≤ν\nu_i(X)\leq\nu

for every finite quotient XX of B\mathcal{B}. This would provide a filling-based form of higher-dimensional expansion that remains meaningful even when cohomology does not vanish. The source gives no resolution, so the conjecture is open.

References

Primary source

Alexander Lubotzky, “Ramanujan Complexes and High Dimensional Expanders”, arXiv:1301.1028 (2013).

Progress summary

Never refreshed

Nothing recorded yet. Refresh searches the literature and the public web for attempts on this problem, and writes the first summary here.

Solutions 0

No solutions have been posted yet.