# First Betti numbers of orbits of Morse functions on surfaces

## TL;DR

This paper investigates the algebraic structure of groups arising from the fundamental groups of Morse function orbits on surfaces, revealing a key property relating their centers and first Betti numbers.

## Contribution

It establishes a novel relation between the ranks of centers and abelianizations of groups generated by products and wreath products, linked to Morse function orbit topology.

## Key findings

- Ranks of the center and abelianization coincide for groups in class G.
- First Betti number of Morse function orbit equals the rank of the group's center.
- Provides algebraic insight into the topology of Morse function orbits on surfaces.

## Abstract

In this article we study algebraic properties of the specific class of groups $\mathcal{G}$ generated by direct products and wreath products. Such class of groups appears in calculation of fundamental groups of orbits of Morse functions on compact manifolds. We prove that for any group $G\in\mathcal{G}$ the ranks of the center $Z(G)$ and the quotient by commutator subgroup $G/[G,G]$ coincide. Moreover, this rank is a first Betti number of the orbit of Morse function.