# Bounds for the rank of the finite part of operator $K$-Theory

## TL;DR

This paper establishes bounds on the finite part of operator K-theory groups for certain groups, linking algebraic properties to geometric invariants, and introduces polynomially full groups where bounds coincide.

## Contribution

It provides new bounds for operator K-theory groups based on group properties and introduces polynomially full groups where these bounds are tight.

## Key findings

- Bounds are derived based on conjugacy classes and torsion elements.
- Polynomially full groups include all virtually nilpotent groups.
- Explicit formulas are given for specific groups like abelian, symmetric, and dihedral groups.

## Abstract

We derive a lower and an upper bound for the rank of the finite part of operator $K$-theory groups of maximal and reduced $C^*$-algebras of finitely generated groups. The lower bound is based on the amount of polynomially growing conjugacy classes of finite order elements in the group. The upper bound is based on the amount of torsion elements in the group. We use the lower bound to give lower bounds for the structure group $S(M)$ and the group of positive scalar curvature metrics $P(M)$ for an oriented manifold $M$. We define a class of groups called "polynomially full groups" for which the upper bound and the lower bound we derive are the same. We show that the class of polynomially full groups contains all virtually nilpotent groups. As example, we give explicit formulas for the ranks of the finite parts of operator $K$-theory groups for the finitely generated abelian groups, the symmetric groups and the dihedral groups.