Automatic strong contraction in higher arithmetic dimension
Determine whether the strong contraction condition δ(Γ) > 1 − [√2(r−1)]⁻¹ holds automatically for every semi-arithmetic Fuchsian group Γ of arithmetic dimension r ≥ 3 that is derived from a quaternion algebra and admits a generalized modular embedding.
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Is eqstrong-contraction (in its sharpened form) automatic for every semi-arithmetic group of arithmetic dimension $r\geq 3$ derived from a quaternion algebra and admitting a generalized modular embedding, as it is for $r\leq 2$?
It therefore remains open whether the class of semi-arithmetic groups of arithmetic dimension $r\geq 3$ satisfying eqstrong-contraction-gen is non-empty.
We have not solved this connection problem, so we still do not exhibit a verified witness, and neither the present techniques nor those of give an explicit handle on $\delta(\Gamma)$ for any specific higher-dimensional example in general.
When $k=2$, eqstrong-contraction-k is the strictly stronger condition $\delta(\Gamma)>1-\bigl(2\sqrt2(r-1)\bigr){-1}$: condition eqstrong-contraction alone does not suffice in this case, and we do not know whether eqstrong-contraction-k can be weakened to eqstrong-contraction when $k=2$ by an argument sharper than the present one.
An analogue for EGMM and generalized modular embeddings would be desirable.