Symmetries of free and right-angled Artin groups
<p>The objects of study in this thesis are automorphism groups of free and right-angled Artin groups. Right-angled Artin groups are defined by a presentation where the only relations are commutators of the generating elements. When there are no relations the right-angled-Artin group is a free...
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| Format: | Thesis |
| Language: | English |
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2012
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| _version_ | 1797090791311540224 |
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| author | Wade, R |
| author2 | Bridson, M |
| author_facet | Bridson, M Wade, R |
| author_sort | Wade, R |
| collection | OXFORD |
| description | <p>The objects of study in this thesis are automorphism groups of free and right-angled Artin groups. Right-angled Artin groups are defined by a presentation where the only relations are commutators of the generating elements. When there are no relations the right-angled-Artin group is a free group and if we take all possible relations we have a free abelian group.</p><p>We show that if no finite index subgroup of a group $G$ contains a normal subgroup that maps onto $mathbb{Z}$, then every homomorphism from $G$ to the outer automorphism group of a free group has finite image. The above criterion is satisfied by SL$_m(mathbb{Z})$ for $m geq 3$ and, more generally, all irreducible lattices in higher-rank, semisimple Lie groups with finite centre.</p><p>Given a right-angled Artin group $A_Gamma$ we find an integer $n$, which may be easily read off from the presentation of $A_G$, such that if $m geq 3$ then SL$_m(mathbb{Z})$ is a subgroup of the outer automorphism group of $A_Gamma$ if and only if $m leq n$. More generally, we find criteria to prevent a group from having a homomorphism to the outer automorphism group of $A_Gamma$ with infinite image, and apply this to a large number of irreducible lattices as above.</p><p>We study the subgroup $IA(A_Gamma)$ of $Aut(A_Gamma)$ that acts trivially on the abelianisation of $A_Gamma$. We show that $IA(A_Gamma)$ is residually torsion-free nilpotent and describe its abelianisation. This is complemented by a survey of previous results concerning the lower central series of $A_Gamma$.</p><p>One of the commonly used generating sets of $Aut(F_n)$ is the set of Whitehead automorphisms. We describe a geometric method for decomposing an element of $Aut(F_n)$ as a product of Whitehead automorphisms via Stallings' folds. We finish with a brief discussion of the action of $Out(F_n)$ on Culler and Vogtmann's Outer Space. In particular we describe translation lengths of elements with regards to the `non-symmetric Lipschitz metric' on Outer Space.</p> |
| first_indexed | 2024-03-07T03:23:47Z |
| format | Thesis |
| id | oxford-uuid:b856e2b5-3689-472b-95c1-71b5748affc9 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T03:23:47Z |
| publishDate | 2012 |
| record_format | dspace |
| spelling | oxford-uuid:b856e2b5-3689-472b-95c1-71b5748affc92022-03-27T04:55:08ZSymmetries of free and right-angled Artin groupsThesishttp://purl.org/coar/resource_type/c_db06uuid:b856e2b5-3689-472b-95c1-71b5748affc9Group theory and generalizations (mathematics)MathematicsEnglishOxford University Research Archive - Valet2012Wade, RBridson, M<p>The objects of study in this thesis are automorphism groups of free and right-angled Artin groups. Right-angled Artin groups are defined by a presentation where the only relations are commutators of the generating elements. When there are no relations the right-angled-Artin group is a free group and if we take all possible relations we have a free abelian group.</p><p>We show that if no finite index subgroup of a group $G$ contains a normal subgroup that maps onto $mathbb{Z}$, then every homomorphism from $G$ to the outer automorphism group of a free group has finite image. The above criterion is satisfied by SL$_m(mathbb{Z})$ for $m geq 3$ and, more generally, all irreducible lattices in higher-rank, semisimple Lie groups with finite centre.</p><p>Given a right-angled Artin group $A_Gamma$ we find an integer $n$, which may be easily read off from the presentation of $A_G$, such that if $m geq 3$ then SL$_m(mathbb{Z})$ is a subgroup of the outer automorphism group of $A_Gamma$ if and only if $m leq n$. More generally, we find criteria to prevent a group from having a homomorphism to the outer automorphism group of $A_Gamma$ with infinite image, and apply this to a large number of irreducible lattices as above.</p><p>We study the subgroup $IA(A_Gamma)$ of $Aut(A_Gamma)$ that acts trivially on the abelianisation of $A_Gamma$. We show that $IA(A_Gamma)$ is residually torsion-free nilpotent and describe its abelianisation. This is complemented by a survey of previous results concerning the lower central series of $A_Gamma$.</p><p>One of the commonly used generating sets of $Aut(F_n)$ is the set of Whitehead automorphisms. We describe a geometric method for decomposing an element of $Aut(F_n)$ as a product of Whitehead automorphisms via Stallings' folds. We finish with a brief discussion of the action of $Out(F_n)$ on Culler and Vogtmann's Outer Space. In particular we describe translation lengths of elements with regards to the `non-symmetric Lipschitz metric' on Outer Space.</p> |
| spellingShingle | Group theory and generalizations (mathematics) Mathematics Wade, R Symmetries of free and right-angled Artin groups |
| title | Symmetries of free and right-angled Artin groups |
| title_full | Symmetries of free and right-angled Artin groups |
| title_fullStr | Symmetries of free and right-angled Artin groups |
| title_full_unstemmed | Symmetries of free and right-angled Artin groups |
| title_short | Symmetries of free and right-angled Artin groups |
| title_sort | symmetries of free and right angled artin groups |
| topic | Group theory and generalizations (mathematics) Mathematics |
| work_keys_str_mv | AT wader symmetriesoffreeandrightangledartingroups |