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A p-group with positive rank gradient

(2012) JOURNAL OF GROUP THEORY. 15(2). p.261-270
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Abstract
We construct, for d >= 2 and epsilon > 0, a d-generated p-group Gamma which, in an asymptotic sense, behaves almost like a d-generated free pro-p-group. We show that a subgroup of index p(n) needs (d - epsilon)p(n) generators, and that the subgroup growth of Gamma satisfies s(p)(n)(Gamma) > s(p)(n)(F-d(p))(1-epsilon), where F-d(p) is the d-generated free pro-p-group. To do this we introduce a new invariant for finitely-generated groups and study some of its basic properties.
Keywords
pro-p-groups, subgroup growth, asymptotic group theory, rank gradient

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Citation

Please use this url to cite or link to this publication:

Chicago
Schlage-Puchta, Jan-Christoph. 2012. “A P-group with Positive Rank Gradient.” Journal of Group Theory 15 (2): 261–270.
APA
Schlage-Puchta, J.-C. (2012). A p-group with positive rank gradient. JOURNAL OF GROUP THEORY, 15(2), 261–270.
Vancouver
1.
Schlage-Puchta J-C. A p-group with positive rank gradient. JOURNAL OF GROUP THEORY. 2012;15(2):261–70.
MLA
Schlage-Puchta, Jan-Christoph. “A P-group with Positive Rank Gradient.” JOURNAL OF GROUP THEORY 15.2 (2012): 261–270. Print.
@article{1989443,
  abstract     = {We construct, for d {\textrangle}= 2 and epsilon {\textrangle} 0, a d-generated p-group Gamma which, in an asymptotic sense, behaves almost like a d-generated free pro-p-group. We show that a subgroup of index p(n) needs (d - epsilon)p(n) generators, and that the subgroup growth of Gamma satisfies s(p)(n)(Gamma) {\textrangle} s(p)(n)(F-d(p))(1-epsilon), where F-d(p) is the d-generated free pro-p-group. To do this we introduce a new invariant for finitely-generated groups and study some of its basic properties.},
  author       = {Schlage-Puchta, Jan-Christoph},
  issn         = {1433-5883},
  journal      = {JOURNAL OF GROUP THEORY},
  keyword      = {pro-p-groups,subgroup growth,asymptotic group theory,rank gradient},
  language     = {eng},
  number       = {2},
  pages        = {261--270},
  title        = {A p-group with positive rank gradient},
  url          = {http://dx.doi.org/10.1515/JGT.2011.101},
  volume       = {15},
  year         = {2012},
}

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