Use this URL to cite or link to this record in EThOS:  https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.580952 
Title:  Free and linear representations of outer automorphism groups of free groups  
Author:  Kielak, Dawid 
ISNI:
0000 0004 2744 3485


Awarding Body:  University of Oxford  
Current Institution:  University of Oxford  
Date of Award:  2012  
Availability of Full Text: 


Abstract:  
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(n1)/2. We further show that the image is finite when n(n1)/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.


Supervisor:  Bridson, Martin R.  Sponsor:  Engineering and Physical Sciences Research Council  
Qualification Name:  Thesis (Ph.D.)  Qualification Level:  Doctoral  
EThOS ID:  uk.bl.ethos.580952  DOI:  Not available  
Keywords:  Group theory and generalizations (mathematics) ; outer automorphisms of free groups ; representations  
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