# Peano dimension of fundamental groups

## TL;DR

This paper introduces the Peano dimension for fundamental groups, generalizes geometric dimension, and proves the conjecture relating it to homotopy dimension for certain continua, addressing a longstanding question.

## Contribution

It defines the Peano dimension for fundamental groups and proves its equality with homotopy dimension for one-dimensional or planar Peano continua.

## Key findings

- Peano dimension generalizes geometric dimension for groups.
- Conjecture confirmed for one-dimensional and planar cases.
- Addresses Cannon and Conner's 2007 question.

## Abstract

We define the Peano dimension for groups arising as fundamental groups, which generalizes the classical definition of geometric dimension of finitely presented groups. We conjecture that the Peano dimension of the fundamental group of a aspherical Peano continuum $X$ is equal to the homotopy dimension of $X$. We prove the conjecture for one-dimensional or planar Peano continua. This answers a question posed by Cannon and Conner in 2007 concerning the homotopy dimension of planar sets.