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We define the Peano dimension for groups arising as fundamental groups, whichgeneralizes the classical definition of geometric dimension of finitelypresented groups. We conjecture that the Peano dimension of the fundamentalgroup 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. Thisanswers a question posed by Cannon and Conner in 2007 concerning the homotopydimension of planar sets.

Peano dimension of fundamental groups