The fundamental groups of one-dimensional wild spaces and the Hawaiian earring

Abstract

Let X X be a one-dimensional space which contains a copy C C of a circle and let it not be semi-locally simply connected at any point on C . C. Then the fundamental group of X X cannot be embeddable into a free σ \sigma -product of n-slender groups, for instance, the fundamental group of the Hawaiian earring. Consequently, any one of the fundamental groups of the Sierpinski gasket, the Sierpinski curve, and the Menger curve is not embeddable into the fundamental group of the Hawaiian earring.