  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.
