Extended F4-buildings and the baby monster
Let Θ be the Baby Monster graph which is the graph on the set of {3, 4}-transpositions in the Baby Monster group B in which two such transpositions are adjacent if their product is a central involution in B. Then Θ is locally the commuting graph of central (root) involutions in 2E6(2). The graph Θ c...
| Main Authors: | , , |
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| Format: | Journal Article |
| Language: | English |
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2011
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| Online Access: | https://hdl.handle.net/10356/92188 http://hdl.handle.net/10220/6871 |
| _version_ | 1811690266802782208 |
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| author | Ivanov, A. A. Pasechnik, Dmitrii V. Shpectorov, Sergey V. |
| author2 | School of Physical and Mathematical Sciences |
| author_facet | School of Physical and Mathematical Sciences Ivanov, A. A. Pasechnik, Dmitrii V. Shpectorov, Sergey V. |
| author_sort | Ivanov, A. A. |
| collection | NTU |
| description | Let Θ be the Baby Monster graph which is the graph on the set of {3, 4}-transpositions in the Baby Monster group B in which two such transpositions are adjacent if their product is a central involution in B. Then Θ is locally the commuting graph of central (root) involutions in 2E6(2). The graph Θ contains a family of cliques of size 120. With respect to the incidence relation defined via inclusion these cliques and the non-empty intersections of two or more of them form a geometry ε(B) with diagram for t = 4 and the action of B on ε(B) is flag-transitive. We show that ε(B) contains subgeome¬tries ε(2E6(2)) and ε(Fi22) with diagrams c.F4(2) and c.F4(1). The stabilizers in B of these subgeometries induce on them flag-transitive actions of 2E6(2) : 2 and Fi22 : 2, respectively. The geometries ε(B), ε(2E6(2)) and ε(Fi22) possess the following properties: (a) any two elements of type 1 are incident to at most one common element of type 2 and (b) three elements of type 1 are pairwise incident to common elements of type 2 if and only if they are incident to a common element of type 5. The paper addresses the classification problem of c.F4(t)-geometries satisfying (a) and (b). We construct three further examples for t = 2 with flag-transitive au¬tomorphism groups isomorphic to 3•2E2(2) : 2, E6(2) : 2 and 226.F4(2) and one for t = 1 with flag-transitive automorphism group 3 • Fi22 : 2. We also study the graph of an arbitrary (non-necessary flag-transitive) c.F4(t)-geometry satisfying (a) and (b) and obtain a complete list of possibilities for the isomorphism type of subgraph induced by the common neighbours of a pair of vertices at distance 2. Finally, we prove that ε(B) is the only c.F4(4)-geometry, satisfying (a) and (b). |
| first_indexed | 2024-10-01T06:01:16Z |
| format | Journal Article |
| id | ntu-10356/92188 |
| institution | Nanyang Technological University |
| language | English |
| last_indexed | 2024-10-01T06:01:16Z |
| publishDate | 2011 |
| record_format | dspace |
| spelling | ntu-10356/921882023-02-28T19:32:23Z Extended F4-buildings and the baby monster Ivanov, A. A. Pasechnik, Dmitrii V. Shpectorov, Sergey V. School of Physical and Mathematical Sciences DRNTU::Science::Mathematics::Geometry Let Θ be the Baby Monster graph which is the graph on the set of {3, 4}-transpositions in the Baby Monster group B in which two such transpositions are adjacent if their product is a central involution in B. Then Θ is locally the commuting graph of central (root) involutions in 2E6(2). The graph Θ contains a family of cliques of size 120. With respect to the incidence relation defined via inclusion these cliques and the non-empty intersections of two or more of them form a geometry ε(B) with diagram for t = 4 and the action of B on ε(B) is flag-transitive. We show that ε(B) contains subgeome¬tries ε(2E6(2)) and ε(Fi22) with diagrams c.F4(2) and c.F4(1). The stabilizers in B of these subgeometries induce on them flag-transitive actions of 2E6(2) : 2 and Fi22 : 2, respectively. The geometries ε(B), ε(2E6(2)) and ε(Fi22) possess the following properties: (a) any two elements of type 1 are incident to at most one common element of type 2 and (b) three elements of type 1 are pairwise incident to common elements of type 2 if and only if they are incident to a common element of type 5. The paper addresses the classification problem of c.F4(t)-geometries satisfying (a) and (b). We construct three further examples for t = 2 with flag-transitive au¬tomorphism groups isomorphic to 3•2E2(2) : 2, E6(2) : 2 and 226.F4(2) and one for t = 1 with flag-transitive automorphism group 3 • Fi22 : 2. We also study the graph of an arbitrary (non-necessary flag-transitive) c.F4(t)-geometry satisfying (a) and (b) and obtain a complete list of possibilities for the isomorphism type of subgraph induced by the common neighbours of a pair of vertices at distance 2. Finally, we prove that ε(B) is the only c.F4(4)-geometry, satisfying (a) and (b). Accepted version 2011-07-06T03:52:30Z 2019-12-06T18:18:55Z 2011-07-06T03:52:30Z 2019-12-06T18:18:55Z 2001 2001 Journal Article Ivanov, A. A., Pasechnik, D. V. & Shpectorov, S. V. (2001). Extended F4-buildings and the Baby Monster. Inventiones Mathematicae, 144(2), 399-433. https://hdl.handle.net/10356/92188 http://hdl.handle.net/10220/6871 10.1007/s002220100143 en Inventiones Mathematicae © 2001 Springer. This is the author created version of a work that has been peer reviewed and accepted for publication by Inventiones Mathematicae, Springer. It incorporates referee’s comments but changes resulting from the publishing process, such as copyediting, structural formatting, may not be reflected in this document. The published version is available at the following DOI: http://dx.doi.org/10.1007/s002220100143. 31 p. application/pdf |
| spellingShingle | DRNTU::Science::Mathematics::Geometry Ivanov, A. A. Pasechnik, Dmitrii V. Shpectorov, Sergey V. Extended F4-buildings and the baby monster |
| title | Extended F4-buildings and the baby monster |
| title_full | Extended F4-buildings and the baby monster |
| title_fullStr | Extended F4-buildings and the baby monster |
| title_full_unstemmed | Extended F4-buildings and the baby monster |
| title_short | Extended F4-buildings and the baby monster |
| title_sort | extended f4 buildings and the baby monster |
| topic | DRNTU::Science::Mathematics::Geometry |
| url | https://hdl.handle.net/10356/92188 http://hdl.handle.net/10220/6871 |
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