Free and linear representations of outer automorphism groups of free groups
<p>For various values of n and m we investigate homomorphisms from Out(F_n) to Out(F_m) and from Out(F_n) to GL_m(K), i.e. the free and linear representations of Out(F_n) respectively.</p> <p>By means of a series of arguments revolving around the representation theory of finite sym...
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| Format: | Thesis |
| Idioma: | English |
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2012
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| _version_ | 1826304505784827904 |
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| author | Kielak, D |
| author2 | Bridson, MR |
| author_facet | Bridson, MR Kielak, D |
| author_sort | Kielak, D |
| collection | OXFORD |
| description | <p>For various values of n and m we investigate homomorphisms from Out(F_n) to Out(F_m) and from Out(F_n) to GL_m(K), i.e. the free and linear representations of Out(F_n) respectively.</p> <p>By means of a series of arguments revolving around the representation theory of finite symmetric subgroups of Out(F_n) we prove that each homomorphism from Out(F_n) to GL_m(K) factors through the natural map p_n from Out(F_n) to GL(H_1(F_n,Z)) = GL_n(Z) whenever n=3, m < 7 and char(K) is not an element of {2,3}, and whenever n>5, m< n(n+1)/2 and char(K) is not an element of {2,3,...,n+1}. We also construct a new infinite family of linear representations of Out(F_n) (where n > 2), which do not factor through p_n. When n is odd these have the smallest dimension among all known representations of Out(F_n) with this property.</p> <p>Using the above results we establish that the image of every homomorphism from Out(F_n) to Out(F_m) is finite whenever n=3 and n < m < 6, and of cardinality at most 2 whenever n > 5 and n < m < n(n-1)/2. We further show that the image is finite when n(n-1)/2 -1 < m < n(n+1)/2.</p> <p>We also consider the structure of normal finite index subgroups of Out(F_n). If N is such then we prove that if the derived subgroup of the intersection of N with the Torelli subgroup T_n < Out(F_n) contains some term of the lower central series of T_n then the abelianisation of N is finite.</p> |
| first_indexed | 2024-03-07T06:18:50Z |
| format | Thesis |
| id | oxford-uuid:f2045fba-1546-4dd3-af9f-7d02c4fc505e |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T06:18:50Z |
| publishDate | 2012 |
| record_format | dspace |
| spelling | oxford-uuid:f2045fba-1546-4dd3-af9f-7d02c4fc505e2022-03-27T12:00:24ZFree and linear representations of outer automorphism groups of free groupsThesishttp://purl.org/coar/resource_type/c_db06uuid:f2045fba-1546-4dd3-af9f-7d02c4fc505eGroup theory and generalizations (mathematics)EnglishOxford University Research Archive - Valet2012Kielak, DBridson, MR<p>For various values of n and m we investigate homomorphisms from Out(F_n) to Out(F_m) and from Out(F_n) to GL_m(K), i.e. the free and linear representations of Out(F_n) respectively.</p> <p>By means of a series of arguments revolving around the representation theory of finite symmetric subgroups of Out(F_n) we prove that each homomorphism from Out(F_n) to GL_m(K) factors through the natural map p_n from Out(F_n) to GL(H_1(F_n,Z)) = GL_n(Z) whenever n=3, m < 7 and char(K) is not an element of {2,3}, and whenever n>5, m< n(n+1)/2 and char(K) is not an element of {2,3,...,n+1}. We also construct a new infinite family of linear representations of Out(F_n) (where n > 2), which do not factor through p_n. When n is odd these have the smallest dimension among all known representations of Out(F_n) with this property.</p> <p>Using the above results we establish that the image of every homomorphism from Out(F_n) to Out(F_m) is finite whenever n=3 and n < m < 6, and of cardinality at most 2 whenever n > 5 and n < m < n(n-1)/2. We further show that the image is finite when n(n-1)/2 -1 < m < n(n+1)/2.</p> <p>We also consider the structure of normal finite index subgroups of Out(F_n). If N is such then we prove that if the derived subgroup of the intersection of N with the Torelli subgroup T_n < Out(F_n) contains some term of the lower central series of T_n then the abelianisation of N is finite.</p> |
| spellingShingle | Group theory and generalizations (mathematics) Kielak, D Free and linear representations of outer automorphism groups of free groups |
| title | Free and linear representations of outer automorphism groups of free groups |
| title_full | Free and linear representations of outer automorphism groups of free groups |
| title_fullStr | Free and linear representations of outer automorphism groups of free groups |
| title_full_unstemmed | Free and linear representations of outer automorphism groups of free groups |
| title_short | Free and linear representations of outer automorphism groups of free groups |
| title_sort | free and linear representations of outer automorphism groups of free groups |
| topic | Group theory and generalizations (mathematics) |
| work_keys_str_mv | AT kielakd freeandlinearrepresentationsofouterautomorphismgroupsoffreegroups |