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The groups (2, π‘š | 𝑛, π‘˜ | 1, π‘ž): Finiteness and homotopy

  • Edward Bennett , Mark Dennis and Martin Edjvet EMAIL logo
From the journal Journal of Group Theory

Abstract

We initiate the study of the groups (l,m∣n,k∣p,q) defined by the presentation ⟨a,b∣al,bm,(a⁒b)n,(ap⁒bq)k⟩. When p=1 and q=m-1, we obtain the group (l,m∣n,k), first systematically studied by Coxeter in 1939. In this paper, we restrict ourselves to the case l=2 and 1n+1k≀12 and give a complete determination as to which of the resulting groups are finite. We also, under certain broadly defined conditions, calculate generating sets for the second homotopy group Ο€2⁒(Z), where 𝑍 is the space formed by attaching 2-cells corresponding to (a⁒b)n and (a⁒bq)k to the wedge sum of the Eilenberg–MacLane spaces 𝑋 and π‘Œ, where Ο€1⁒(X)β‰…C2 and Ο€1⁒(Y)β‰…Cm; in particular, Ο€1(Z)β‰…(2,m∣n,k∣1,q).

  1. Communicated by: Christopher W. Parker

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Received: 2020-01-21
Revised: 2020-11-23
Published Online: 2020-12-22
Published in Print: 2021-05-01

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