Let $\Sigma _{g,p}$ be the genus–g oriented surface with p punctures, with either g > 0 or p > 3. We show that $MCG(\Sigma _{g,p})/DT$ is acylindrically hyperbolic where DT is the normal subgroup of the mapping class group $MCG(\Sigma _{g,p})$ generated by $K^{th}$ powers of Dehn twists about curves in $\Sigma _{g,p}$ for suitable K.Moreover, we show that in low complexity $MCG(\Sigma _{g,p})/DT$ is in fact hyperbolic. In particular, for 3g − 3 + p ⩽ 2, we show that the mapping class group $MCG(\Sigma _{g,p})$ is fully residually non-elementary hyperbolic and admits an affine isometric action with unbounded orbits on some $L^q$ space. Moreover, if every hyperbolic group is residually finite, then every convex-cocompact subgroup of $MCG(\Sigma _{g,p})$ is separable.The aforementioned results follow from general theorems about composite rotating families, in the sense of [13], that come from a collection of subgroups of vertex stabilizers for the action of a group G on a hyperbolic graph X. We give conditions ensuring that the graph X/N is again hyperbolic and various properties of the action of G on X persist for the action of G/N on X/N.

中文翻译：

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德恩填充德恩曲折

让$\Sigma _{g,p}$成为一个属——G定向表面p穿刺，无论是G> 0 或p> 3. 我们证明了$MCG(\Sigma _{g,p})/DT$是圆柱形双曲线，其中DT是映射类群的正规子群$MCG(\Sigma _{g,p})$由产生$K^{th}$德恩的力量在曲线上扭曲$\Sigma _{g,p}$适合ķ.此外，我们表明，在低复杂度$MCG(\Sigma _{g,p})/DT$实际上是双曲线的。特别是对于 3G− 3 +p⩽ 2，我们证明了映射类组$MCG(\Sigma _{g,p})$是完全残差非初等双曲的，并且允许仿射等距作用在某些上具有无界轨道$L^q$空间。此外，如果每个双曲群都是残差有限的，那么$MCG(\Sigma _{g,p})$是可分离的。上述结果来自关于复合旋转族的一般定理，在 [13] 的意义上，这些定理来自一组顶点稳定器的子组，用于组的作用G在双曲线图上X. 我们给出条件确保图X/ñ又是双曲线的和作用的各种性质G在X坚持采取行动G/ñ在X/ñ.