On the Dehn functions of K\"ahler groups

TL;DR

This paper investigates the possible Dehn functions of K"ahler groups, providing examples with various growth rates and showing that certain curvature conditions do not guarantee quadratic Dehn functions.

Contribution

It demonstrates the existence of K"ahler groups with Dehn functions bounded between cubic and sixth power, challenging assumptions about curvature and Dehn function behavior.

Findings

Existence of K"ahler groups with linear, quadratic, and exponential Dehn functions.

Construction of a K"ahler group with Dehn function between n^3 and n^6.

Non-positive holomorphic bisectional curvature does not imply quadratic Dehn function.

Abstract

We address the problem of which functions can arise as Dehn functions of K\"ahler groups. We explain why there are examples of K\"ahler groups with linear, quadratic, and exponential Dehn function. We then proceed to show that there is an example of a K\"ahler group which has Dehn function bounded below by a cubic function and above by $n^6$. As a consequence we obtain that for a compact K\"ahler manifold having non-positive holomorphic bisectional curvature does not imply having quadratic Dehn function.