Abstract. Continuing Cavicchioli, Repovš, and Spaggiari's investigations into the cyclic presentations hx 1 ; : : : ; x n j x i x i Ck x i Cl D 1 .1 Ä i Ä n/i we determine when they are aspherical and when they define finite groups; in these cases we describe the groups' structures. In many cases we show that if the group is infinite then it contains a non-abelian free subgroup. Mathematics Subject Classification (2010). 20F05, 20F06.
Abstract. The Cavicchioli-Hegenbarth-Repovš generalized Fibonacci groups are defined by the presentations G n ðm; kÞ ¼ hx 1 ; . . . ; x n j x i x iþm ¼ x iþk ð1 c i c nÞi. These cyclically presented groups generalize Conway's Fibonacci groups and the Sieradski groups. Building on a theorem of Bardakov and Vesnin we classify the aspherical presentations G n ðm; kÞ. We determine when G n ðm; kÞ has infinite abelianization and provide su‰cient conditions for G n ðm; kÞ to be perfect. We conjecture that these are also necessary conditions. Combined with our asphericity theorem, a proof of this conjecture would imply a classification of the finite CavicchioliHegenbarth-Repovš groups.
We study a class of Labelled Oriented Graph (LOG) group where the underlying graph is a tadpole graph. We show that such a group is the natural HNN extension of a cyclically presented group and investigate the relationship between the LOG group and the cyclically presented group. We relate the second homotopy groups of their presentations and show that hyperbolicity of the cyclically presented group implies solvability of the conjugacy problem for the LOG group. In the case where the label on the tail of the LOG spells a positive word in the vertices in the circuit we show that the LOGs and groups coincide with those considered by Szczepański and Vesnin. We obtain new presentations for these cyclically presented groups and show that the groups of Fibonacci type introduced by Johnson and Mawdesley are of this form. These groups generalize the Fibonacci groups and the Sieradski groups and have been studied by various authors. We continue these investigations, using small cancellation and curvature methods to obtain results on hyperbolicity, automaticity, SQ-universality, and solvability of decision problems.
In gratitude to Stephen J. Pride for his friendship and mentorship over the years. AbstractThe concept of asphericity for relative group presentations was introduced twenty five years ago. Since then, the subject has advanced and detailed asphericity classifications have been obtained for various families of one-relator relative presentations. Through this work the definition of asphericity has evolved and new applications have emerged.In this article we bring together key results on relative asphericity, update them, and exhibit them under a single set of definitions and terminology. We describe consequences of asphericity and present techniques for proving asphericity and for proving non-asphericity. We give a detailed survey of results concerning one-relator relative presentations where the relator has free product length four.
We study the cyclic presentations with relators of the form xixi+mx −1 i+k 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.
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