On the proofs of Leighton's Graph Covering Theorem, a notion dual to commensurability, and normal virtual retracts
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909779547389952 |
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| author | Touikan, Nicholas Minasyan, Ashot |
| author_facet | Touikan, Nicholas Minasyan, Ashot |
| contents | Leighton's Graph Covering Theorem states that if two finite graphs have the same universal covering tree, then they also have a common finite degree cover. Bass and Kulkarni gave an alternative proof of this fact using tree lattices. We give an example of two graphs that admit a common finite cover which can not be obtained using tree lattice techniques. If two groups embed as finite index subgroups, we say they are co-commensurable. Our example comes from an explicit commensuration that cannot be induced by a co-commensuration. Next we state and prove a general theorem that gives necessary and sufficient conditions for when a commensuration can be induced by a co-commensuration. The developed machinery is then used to show that normal virtual retracts are virtual direct summands, answering a question of Merladet and Minasyan. In an appendix, applications to commensurating graphs of groups, biautomaticity, and hereditary conjugacy separability are given. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_01839 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the proofs of Leighton's Graph Covering Theorem, a notion dual to commensurability, and normal virtual retracts Touikan, Nicholas Minasyan, Ashot Group Theory 20F65, 05C25, 05C70, 20E06 Leighton's Graph Covering Theorem states that if two finite graphs have the same universal covering tree, then they also have a common finite degree cover. Bass and Kulkarni gave an alternative proof of this fact using tree lattices. We give an example of two graphs that admit a common finite cover which can not be obtained using tree lattice techniques. If two groups embed as finite index subgroups, we say they are co-commensurable. Our example comes from an explicit commensuration that cannot be induced by a co-commensuration. Next we state and prove a general theorem that gives necessary and sufficient conditions for when a commensuration can be induced by a co-commensuration. The developed machinery is then used to show that normal virtual retracts are virtual direct summands, answering a question of Merladet and Minasyan. In an appendix, applications to commensurating graphs of groups, biautomaticity, and hereditary conjugacy separability are given. |
| title | On the proofs of Leighton's Graph Covering Theorem, a notion dual to commensurability, and normal virtual retracts |
| topic | Group Theory 20F65, 05C25, 05C70, 20E06 |
| url | https://arxiv.org/abs/2507.01839 |