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A Geometric Study of Commutator Subgroups


Zhuang, Dongping (2009) A Geometric Study of Commutator Subgroups. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/J566-2537.


Let G be a group and G' its commutator subgroup. Commutator length (cl) and stable commutator length (scl) are naturally defined concepts for elements of G'. We study cl and scl for two classes of groups. First, we compute scl in generalized Thompson's groups and their central extensions. As a consequence, we find examples of finitely presented groups in which scl takes irrational (in fact, transcendental) values. Second, we study large scale geometry of the Cayley graph of a commutator subgroup with respect to the canonical generating set of all commutators. When G is a non-elementary hyperbolic group, we prove that, for any n, there exists a quasi-isometrically embedded, dimension n integral lattice in this graph. Thus this graph is not hyperbolic, has infinite asymptotic dimension, and has only one end. For a general finitely presented group, we show that this graph is large scale simply connected.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:commutator length and stable commutator length; generalized Thompson's group; large scale geometry
Degree Grantor:California Institute of Technology
Division:Physics, Mathematics and Astronomy
Major Option:Mathematics
Awards:Scott Russell Johnson Prize for Excellence in Graduate Study in Mathematics, 2008.
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Calegari, Danny C.
Thesis Committee:
  • Calegari, Danny C. (chair)
  • Day, Matthew B.
  • Aschbacher, Michael
  • Graber, Thomas B.
Defense Date:12 May 2009
Record Number:CaltechETD:etd-05152009-115934
Persistent URL:
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:1827
Deposited By: Imported from ETD-db
Deposited On:29 May 2009
Last Modified:26 Nov 2019 20:23

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