Fibonacci type presentations and 3-manifolds

We study the cyclic presentations with relators of the form xixi+mxi+k−1 and the groups they define. These “groups of Fibonacci type” were introduced by Johnson and Mawdesley and they generalise 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 RepovÅ¡. 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.