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. 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. 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. 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 $A_Gamma$. 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.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:572599 |
| Date | January 2012 |
| Creators | Wade, Richard D. |
| Contributors | Bridson, Martin R. |
| Publisher | University of Oxford |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | http://ora.ox.ac.uk/objects/uuid:b856e2b5-3689-472b-95c1-71b5748affc9 |
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