Use this URL to cite or link to this record in EThOS:  https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.729914 
Title:  Word fibres in finite pgroups and prop groups  
Author:  IniguezGoizueta, Ainhoa 
ISNI:
0000 0004 6498 8986


Awarding Body:  University of Oxford  
Current Institution:  University of Oxford  
Date of Award:  2016  
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Abstract:  
Given a group word w in k variables, a group GM and g ∈ G, we consider the set S_{w}(g) of ktuples (g_{1},..., g_{k}) ∈ G^{(k)} such that w(g_{1},..., g_{k}) = g and when G is finite, the size of S_{w}(g), N_{w}(g). N. Amit conjectured that for any finite nilpotent group G and any word in k variables, N_{w}(1) ≥ G^{k1}. In this thesis we first prove Amit's conjecture for finite groups of nilpotency class 2. This was independently proved by Levy in [1]. More generally, we study the class functions N_{w} for this class of groups and show that the inequality can be improved to N_{w}(1) ≥ G^{k}/G_{w} (G_{w} is the set of wvalues in G) if G has odd order. This last result is explained by the fact that the functions N_{w} are characters of G in this case. For groups of even order, all that can be said is that N_{w} is a generalized character, something that is false in general for groups of nilpotency class greater than 2. We characterize group theoretically when N_{x}^{n} is a character if G is a 2group of nilpotency class 2. We also address the (much harder) problem of studying if N_{w}(g) ≥ G^{k1} for g ∈ G_{w}, proving that this is the case for the free pgroups of nilpotency class 2 and exponent p. Finally, we look at the analogous problem for finitely generated prop groups. Let G be a finitely generated prop group and {G_{n}} some filtration. We define the dimension of a closed subset H ⊆ G as
 ^{lim inf}_{n→∞}  log_{p}  HG^{(k)}_{n}/ G^{(k)}_{n} log_{p}  (G / G_{n})^{(k)}   ·  
Supervisor:  Segal, Dan ; Bridson, Martin ; Camina, Rachel  Sponsor:  Not available  
Qualification Name:  Thesis (Ph.D.)  Qualification Level:  Doctoral  
EThOS ID:  uk.bl.ethos.729914  DOI:  Not available  
Keywords:  Word fibres ; Finite groups ; pgroups ; fibres  
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