We show that Cannon–Thurston maps exist for degenerate free groups without parabolics, that is, for handlebody groups. Combining these techniques with earlier work proving the existence of Cannon–Thurston maps for surface groups, we show that Cannon–Thurston maps exist for arbitrary finitely generated Kleinian groups without parabolics, proving conjectures of Thurston and McMullen. We also show that point pre-images under Cannon–Thurston maps for degenerate free groups without parabolics correspond to endpoints of leaves of an ending lamination in the Masur domain, whenever a point has more than one pre-image. This proves a conjecture of Otal. We also prove a similar result for point pre-images under Cannon–Thurston maps for arbitrary finitely generated Kleinian groups without parabolics.

中文翻译：

---

克莱尼集团的坎农-瑟斯顿地图

我们证明了 Cannon-Thurston 映射存在于没有抛物线的退化自由群，即手柄体群。将这些技术与证明表面群存在 Cannon-Thurston 映射的早期工作相结合，我们证明 Cannon-Thurston 映射存在于任意有限生成的没有抛物线的 Kleinian 群，证明了 Thurston 和 McMullen 的猜想。我们还展示了 Cannon-Thurston 映射下没有抛物线的退化自由群的点原像对应于 Masur 域中末端层压的叶子的端点，只要一个点具有多个原像。这证明了奥塔尔的一个猜想。对于没有抛物线的任意有限生成的 Kleinian 群，我们还证明了 Cannon-Thurston 映射下的点原像的类似结果。