标题： 关于分层双曲空间边界的连通性 显示英文标题

标题： On the connectedness of the boundary of hierarchically hyperbolic spaces

摘要： 我们证明，在一个温和的假设下，单端的紧致度量空间的任何可度量化的紧化都是连通的。 作为推论，我们得出结论，由Durham--Hagen--Sisto引入的单端分层双曲空间的边界是连通的。 此外，我们证明分层双曲群的边界的连通性等价于该群的单端性。 作为应用，我们证明如果对于$n\geq 2$，$G_1=A_1\ast\dots\ast A_n$和$G_2=B_1\ast\dots\ast B_n$是单端分层双曲群的自由积，则$G_1$的边界与$G_2$的边界同胚当且仅当$A_i$的边界与$B_i$的边界同胚对于$1\leq i\leq n$。 显示英文摘要

摘要： We prove that, under a mild assumption, any metrizable compactification of a one-ended proper geodesic metric space is connected. As a consequence, we deduce that the boundary, introduced by Durham--Hagen--Sisto, of a one-ended hierarchically hyperbolic space is connected. Moreover, we prove that the connectedness of the boundary of a hierarchically hyperbolic group is equivalent to the one-endedness of the group. As an application, we show that if, for $n\geq 2$, $G_1=A_1\ast\dots\ast A_n$ and $G_2=B_1\ast\dots\ast B_n$ are free products of one-ended hierarchically hyperbolic groups, then the boundary of $G_1$ is homeomorphic to the boundary of $G_2$ if and only if the boundary of $A_i$ is homeomorphic to the boundary of $B_i$ for $1\leq i\leq n$.