We give an effective upper bound, for certain arithmetic hyperbolic 3-manifold groups obtained from a quadratic form construction, on the minimal index of a subgroup that embeds in a fixed 6-dimensional right-angled reflection group, stabilizing a totally geodesic subspace. In particular, for manifold groups in any fixed commensurability class we show that the index of such a subgroup is asymptotically smaller than any fractional power of the volume of the manifold. We also give effective bounds on the geodesic residual finiteness growths of closed hyperbolic manifolds that totally geodesically immerse in non-compact right-angled reflection orbifolds, extending work of the third author from the compact case. The first result gives examples to which the second applies, and for these we give explicit bounds on geodesic residual finiteness growth.

中文翻译：

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一些算术双曲三流形的有效虚和残差性质

我们给出了一个有效的上限，对于从二次形式构造获得的某些算术双曲 3 流形群，在嵌入固定 6 维直角反射群的子群的最小索引上，稳定一个完全测地线子空间。特别是，对于任何固定可公度类中的流形群，我们表明这种子群的指数渐近地小于流形体积的任何分数幂。我们还给出了闭合双曲流形的测地剩余有限度增长的有效边界，这些流形完全测地浸入非紧直角反射轨道，扩展了第三作者的工作。第一个结果给出了第二个适用的例子，对于这些，我们给出了测地线残差有限性增长的明确界限。