This paper is a continuation of our previous work with Margalit where we studied group actions on projection complexes. In that paper, we demonstrated sufficient conditions so that the normal closure of a family of subgroups of vertex stabilizers is a free product of certain conjugates of these subgroups. In this paper, we study both the quotient of the projection complex by this normal subgroup and the action of the quotient group on the quotient of the projection complex. We show that under certain conditions the quotient complex is $\delta$-hyperbolic. Additionally, under certain circumstances, we show that if the original action on the projection complex was a non-elementary WPD action, then so is the action of the quotient group on the quotient of the projection complex. This implies that the quotient group is acylindrically hyperbolic.

中文翻译：

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投影复数的双曲商

这篇论文是我们之前与 Margalit 合作的延续，我们研究了投影复合体上的群体行动。在那篇论文中，我们证明了充分的条件，使得顶点稳定器子群的正常闭包是这些子群的某些共轭的自由乘积。在本文中，我们研究了投影复数与这个正规子群的商以及商群对投影复数商的作用。我们证明，在某些条件下，商复数是 $\delta$-双曲线。此外，在某些情况下，我们表明，如果对投影复合体的原始作用是非基本 WPD 作用，那么商群对投影复合体商的作用也是如此。这意味着商群是非圆柱双曲线的。