# The slope of surfaces with Albanese dimension one

## TL;DR

This paper constructs new examples of surfaces with Albanese dimension one, showing that their slopes are dense in the interval [8,9], extending previous results on the possible slope values for such surfaces.

## Contribution

It provides a novel construction of surfaces with Albanese dimension one, filling the gap in the slope interval [8,9] using ramified double coverings of cyclic covers.

## Key findings

- Surfaces with Albanese dimension one can have slopes dense in [8,9].
- New construction involves ramified double coverings of cyclic covers of the Cartwright-Steger surface.
- Extends the known range of slopes for these surfaces.

## Abstract

Mendes Lopes and Pardini showed that minimal general type surfaces of Albanese dimension one have slopes $K^2/\chi$ dense in the interval $[2,8]$. This result was completed to cover the admissible interval $[2,9]$ by Roulleau and Urzua, who proved that surfaces with fundamental group equal to that of any curve of genus $g \geq 1$ (in particular, having Albanese dimension one) give a set of slopes dense in $[6,9]$. In this note we provide a second construction that complements that of Mendes Lopes-Pardini, to recast a dense set of slopes in $[8,9]$ for surfaces of Albanese dimension one. These surfaces arise as ramified double coverings of cyclic covers of the Cartwright-Steger surface.