On the ranks of the algebraic $K$-Theory of hyperbolic groups

TL;DR

This paper proves that the algebraic K-theory groups of group rings for hyperbolic groups have finite rank across all degrees, providing explicit formulas for certain classes.

Contribution

It establishes the finiteness of ranks of algebraic K-theory groups for hyperbolic groups and derives explicit formulas for specific cases.

Findings

K_n(Z[G]) has finite rank for all n in Z

Explicit formulas for ranks in certain group classes

Advances understanding of algebraic K-theory of hyperbolic groups

Abstract

Let $G$ be a word hyperbolic group. We prove that the algebraic $K$-theory groups of $\dbZ [G]$, $K_n(\dbZ[G])$, have finite rank for all $n\in \dbZ$. For a few classes of groups, we give explicit formulas for the ranks of the algebraic $K$-theory groups of their group rings.