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| Main Author: | |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2509.00321 |
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| _version_ | 1866918132952596480 |
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| author | Tomar, Ravi |
| author_facet | Tomar, Ravi |
| contents | 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 $1\leq i\leq n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_00321 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the connectedness of the boundary of hierarchically hyperbolic spaces Tomar, Ravi Group Theory 20F65, 20F67 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 $1\leq i\leq n$. |
| title | On the connectedness of the boundary of hierarchically hyperbolic spaces |
| topic | Group Theory 20F65, 20F67 |
| url | https://arxiv.org/abs/2509.00321 |