Let $H<{\mathrm{PSL}}_2(\mathbb{Z})$ be a finite index normal subgroup which is contained in a principal congruence subgroup, and let $\mathrm{\Phi }(H)\ne H$ denote a term of the lower central series or the derived series of $H$. In this paper, we prove that the commensurator of $\mathrm{\Phi }(H)$ in ${\mathrm{PSL}}_2(\mathbb{R})$ is discrete. We thus obtain a natural family of thin subgroups of ${\mathrm{PSL}}_2(\mathbb{R})$ whose commensurators are discrete, establishing some cases of a conjecture of Shalom.

中文翻译：

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换向器、公称器和 PSL2(Z)

让 $H<{\mathrm{PSL}}_2(\mathbb{Z})$ 是包含在主同余子群中的有限索引正规子群，令 $\mathrm{\Phi }(H)\ne H$ 表示下中心级数或派生级数的项 $H$. 在本文中，我们证明了$\mathrm{\Phi }(H)$ 在 ${\mathrm{PSL}}_2(\mathbb{电阻})$是离散的。因此，我们获得了一个自然的瘦子群家族${\mathrm{PSL}}_2(\mathbb{电阻})$ 其对称子是离散的，建立了沙洛姆猜想的一些情况。