Weak commutativity, virtually nilpotent groups, and Dehn functions
The group <i>X</i>(<i>G</i>) is obtained from <i>G</i>∗<i>G</i> by forcing each element <i>g</i> in the first free factor to commute with the copy of <i>g</i> in the second free factor. We make significant additions to the list...
| मुख्य लेखकों: | , |
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| स्वरूप: | Journal article |
| भाषा: | English |
| प्रकाशित: |
EMS Press
2023
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| _version_ | 1826313049633456128 |
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| author | Bridson, MR Kochloukova, DH |
| author_facet | Bridson, MR Kochloukova, DH |
| author_sort | Bridson, MR |
| collection | OXFORD |
| description | The group <i>X</i>(<i>G</i>) is obtained from <i>G</i>∗<i>G</i> by forcing each element <i>g</i> in the first free factor to commute with the copy of <i>g</i> in the second free factor. We make significant additions to the list of properties that the functor <i>X</i> is known to preserve. We also investigate the geometry and complexity of the word problem for <i>X</i>(<i>G</i>). Subtle features of <i>X</i>(<i>G</i>) are encoded in a normal abelian subgroup <i>W</i><<i>X</i>(<i>G</i>) that is a module over <i>ZQ</i>, where <i>Q</i>=<i>H</i><sub>1</sub> (<i>G</i>,<i>Z</i>). We establish a structural result for this module and illustrate its utility by proving that <i>X</i> preserves virtual nilpotence, the Engel condition, and growth type – polynomial, exponential, or intermediate. We also use it to establish isoperimetric inequalities for <i>X</i>(<i>G</i>) when <i>G</i> lies in a class that includes Thompson's group <i>F</i> and all non-fibred Kähler groups. The word problem is soluble in <i>X</i>(<i>G</i>) if and only if it is soluble in <i>G</i>. The Dehn function of <i>X</i>(<i>G</i>) is bounded below by a cubic polynomial if <i>G</i> maps onto a non-abelian free group. |
| first_indexed | 2024-03-07T08:25:53Z |
| format | Journal article |
| id | oxford-uuid:3a4d8db1-793d-494c-a3d6-140634bc32cf |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-09-25T04:04:53Z |
| publishDate | 2023 |
| publisher | EMS Press |
| record_format | dspace |
| spelling | oxford-uuid:3a4d8db1-793d-494c-a3d6-140634bc32cf2024-05-22T16:43:25ZWeak commutativity, virtually nilpotent groups, and Dehn functionsJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:3a4d8db1-793d-494c-a3d6-140634bc32cfEnglishSymplectic ElementsEMS Press2023Bridson, MRKochloukova, DHThe group <i>X</i>(<i>G</i>) is obtained from <i>G</i>∗<i>G</i> by forcing each element <i>g</i> in the first free factor to commute with the copy of <i>g</i> in the second free factor. We make significant additions to the list of properties that the functor <i>X</i> is known to preserve. We also investigate the geometry and complexity of the word problem for <i>X</i>(<i>G</i>). Subtle features of <i>X</i>(<i>G</i>) are encoded in a normal abelian subgroup <i>W</i><<i>X</i>(<i>G</i>) that is a module over <i>ZQ</i>, where <i>Q</i>=<i>H</i><sub>1</sub> (<i>G</i>,<i>Z</i>). We establish a structural result for this module and illustrate its utility by proving that <i>X</i> preserves virtual nilpotence, the Engel condition, and growth type – polynomial, exponential, or intermediate. We also use it to establish isoperimetric inequalities for <i>X</i>(<i>G</i>) when <i>G</i> lies in a class that includes Thompson's group <i>F</i> and all non-fibred Kähler groups. The word problem is soluble in <i>X</i>(<i>G</i>) if and only if it is soluble in <i>G</i>. The Dehn function of <i>X</i>(<i>G</i>) is bounded below by a cubic polynomial if <i>G</i> maps onto a non-abelian free group. |
| spellingShingle | Bridson, MR Kochloukova, DH Weak commutativity, virtually nilpotent groups, and Dehn functions |
| title | Weak commutativity, virtually nilpotent groups, and Dehn functions |
| title_full | Weak commutativity, virtually nilpotent groups, and Dehn functions |
| title_fullStr | Weak commutativity, virtually nilpotent groups, and Dehn functions |
| title_full_unstemmed | Weak commutativity, virtually nilpotent groups, and Dehn functions |
| title_short | Weak commutativity, virtually nilpotent groups, and Dehn functions |
| title_sort | weak commutativity virtually nilpotent groups and dehn functions |
| work_keys_str_mv | AT bridsonmr weakcommutativityvirtuallynilpotentgroupsanddehnfunctions AT kochloukovadh weakcommutativityvirtuallynilpotentgroupsanddehnfunctions |