论文标题
$ \ mathrm {sl} _2（\ Mathbb {r}）$和某些Ramanujan图的小直径和算术晶格的发电机
Small diameters and generators for arithmetic lattices in $\mathrm{SL}_2(\mathbb{R})$ and certain Ramanujan graphs
论文作者
论文摘要
我们表明，$ \ mathrm {sl} _ {2}（\ Mathbb {r}）$中的算术晶格是源于在$ \ mathbb {q} $上的$ \ mathbb {q} $上的eichler订单的适当单位，允许使用$ \ mathbb {q} $。特别是，如果商是共同的，我们会在直径上给出界限。因此，我们表明这些晶格允许小发电机。我们的技术还适用于确定的四元组代数，在该代数中，我们在某些Ramanujan图的直径上显示了Ramanujan-Sprength Bends在不使用Ramanujan结合的情况下。
We show that arithmetic lattices in $\mathrm{SL}_{2}(\mathbb{R})$, stemming from the proper units of an Eichler order in an indefinite quaternion algebra over $\mathbb{Q}$, admit a `small' covering set. In particular, we give bounds on the diameter if the quotient space is co-compact. Consequently, we show that these lattices admit small generators. Our techniques also apply to definite quaternion algebras where we show Ramanujan-strength bounds on the diameter of certain Ramanujan graphs without the use of the Ramanujan bound.
