Fibonacci type presentations and 3-manifolds

TL;DR

This paper classifies cyclic groups of Fibonacci type as fundamental groups of 3-manifolds, identifies which presentations serve as spines of 3-manifolds, and explores alternative presentations for even n.

Contribution

It provides a complete classification of Fibonacci type groups as 3-manifold groups and characterizes presentations that are spines of 3-manifolds, answering an open question.

Findings

Only Fibonacci, Sieradski, and cyclic groups are 3-manifold groups.

Classified presentations that are spines of 3-manifolds.

Alternative presentations for even n also serve as spines.

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 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 Repov\v{s}. 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.