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Free and linear representations of outer automorphism groups of free groups

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. 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. 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. 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.

Identiferoai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:580952
Date January 2012
CreatorsKielak, Dawid
ContributorsBridson, Martin R.
PublisherUniversity of Oxford
Source SetsEthos UK
Detected LanguageEnglish
TypeElectronic Thesis or Dissertation
Sourcehttp://ora.ox.ac.uk/objects/uuid:f2045fba-1546-4dd3-af9f-7d02c4fc505e

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