Fibonacci type presentations and 3-manifolds

James Howie, Gerald Williams

Research output: Contribution to journalArticle

4 Citations (Scopus)
32 Downloads (Pure)

Abstract

We study the cyclic presentations with relators of the form $x_ix_{i+m}x_{i+k}^{-1}$ and the groups they define. These "groups of We study the cyclic presentations with relators of the form View the MathML source and the groups they define. These “groups of Fibonacci type” were introduced by Johnson and Mawdesley and they generalize the Fibonacci groups F(2,n) and the Sieradski groups S(2,n). With the exception of two groups, we classify when these groups are fundamental groups of 3-manifolds, and it turns out that only Fibonacci, Sieradski, and cyclic groups arise. Using this classification, we completely classify the presentations that are spines of 3-manifolds, answering a question of Cavicchioli, Hegenbarth, and Repovš. When n is even the groups F(2,n),S(2,n) admit alternative cyclic presentations on n/2 generators. We show that these alternative presentations also arise as spines of 3-manifolds.
Original languageEnglish
Pages (from-to)24–34
Number of pages11
JournalTopology and its Applications
Volume215
Early online date4 Nov 2016
DOIs
Publication statusPublished - 1 Jan 2017

Keywords

  • math.GT
  • 20F05, 57M05 (primary), 57M50 (secondary)

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