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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2404.15459 |
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| _version_ | 1866910421157412864 |
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| author | Balasubramanya, Sahana Burkhalter, Georgia Niebler, Rachel Shapiro, Roberta |
| author_facet | Balasubramanya, Sahana Burkhalter, Georgia Niebler, Rachel Shapiro, Roberta |
| contents | A group has Property $\mathrm{(NL)}$ if it does not admit a loxodromic element in any hyperbolic action. In other words, a group with this property is inaccessible for study from the perspective of hyperbolic actions. This property was introduced by Balasubramanya, Fournier-Facio and Genevois, who initiated the study of this property. We expand on this research by studying Property $\mathrm{(NL)}$ in Coxeter groups, a class of groups that are defined by an underlying graph. One of our main results show that a right-angled Coxeter group (RACG) has Property $\mathrm{(NL)}$ if and only if its defining graph is complete. We then move beyond the right-angled case to show that if a defining graph is disconnected, its corresponding Coxeter group does not have Property $\mathrm{(NL)}$. Lastly, we classify which triangle groups (Coxeter groups with three generators) have Property $\mathrm{(NL)}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_15459 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Property $\mathrm{(NL)}$ in Coexeter groups Balasubramanya, Sahana Burkhalter, Georgia Niebler, Rachel Shapiro, Roberta Group Theory General Topology 20F65 A group has Property $\mathrm{(NL)}$ if it does not admit a loxodromic element in any hyperbolic action. In other words, a group with this property is inaccessible for study from the perspective of hyperbolic actions. This property was introduced by Balasubramanya, Fournier-Facio and Genevois, who initiated the study of this property. We expand on this research by studying Property $\mathrm{(NL)}$ in Coxeter groups, a class of groups that are defined by an underlying graph. One of our main results show that a right-angled Coxeter group (RACG) has Property $\mathrm{(NL)}$ if and only if its defining graph is complete. We then move beyond the right-angled case to show that if a defining graph is disconnected, its corresponding Coxeter group does not have Property $\mathrm{(NL)}$. Lastly, we classify which triangle groups (Coxeter groups with three generators) have Property $\mathrm{(NL)}$. |
| title | Property $\mathrm{(NL)}$ in Coexeter groups |
| topic | Group Theory General Topology 20F65 |
| url | https://arxiv.org/abs/2404.15459 |