Short Geodesic Conjecture for arithmetic hyperbolic orbifolds

Let an arithmetic hyperbolic orbifold of the first type be a quotient of hyperbolic space by an arithmetic lattice of the first type. For a hyperbolic element γ\gamma, write ℓ(γ)\ell(\gamma) for the length of its associated closed geodesic. Short Geodesic Conjecture. There exists a positive universal lower bound on the length of geodesics in an arithmetic hyperbolic orbifold of the first type. Through the relationship λ=eℓ(γ)\lambda=e^{\ell(\gamma)} between geodesic lengths and Salem numbers, this is presented as a reformulation of Lehmer's conjecture. The source does not provide evidence resolving the conjecture.

References

Primary source

Michelle Chu, Plinio G. P. Murillo, Otto Romero and Lola Thompson, “Counting Salem numbers arising from arithmetic hyperbolic orbifolds”, arXiv:2508.08003 (2026).

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.