Word calculus in the fundamental group of the Menger curve

• Published in 2013
• Added on 2020-05-02

The fundamental group of the Menger universal curve is uncountable and not free, although all of its finitely generated subgroups are free. It contains an isomorphic copy of the fundamental group of every one-dimensional separable metric space and an isomorphic copy of the fundamental group of every planar Peano continuum. We give an explicit and systematic combinatorial description of the fundamental group of the Menger universal curve and its generalized Cayley graph in terms of word sequences. The word calculus, which requires only two letters and their inverses, is based on Pasynkov's partial topological product representation and can be expressed in terms of a variation on the classical puzzle known as the Towers of Hanoi.

Links

• http://arxiv.org/abs/1310.7968v1
• http://arxiv.org/pdf/1310.7968v1

Other information

BibTeX entry

@article{WordcalculusinthefundamentalgroupoftheMengercurve,
	key = {WordcalculusinthefundamentalgroupoftheMengercurve},
	type = {article},
	title = {Word calculus in the fundamental group of the Menger curve},
	author = {Hanspeter Fischer and Andreas Zastrow},
	abstract = {The fundamental group of the Menger universal curve is uncountable and not
free, although all of its finitely generated subgroups are free. It contains an
isomorphic copy of the fundamental group of every one-dimensional separable
metric space and an isomorphic copy of the fundamental group of every planar
Peano continuum. We give an explicit and systematic combinatorial description
of the fundamental group of the Menger universal curve and its generalized
Cayley graph in terms of word sequences. The word calculus, which requires only
two letters and their inverses, is based on Pasynkov's partial topological
product representation and can be expressed in terms of a variation on the
classical puzzle known as the Towers of Hanoi.},
	comment = {},
	date_added = {2020-05-02},
	date_published = {2013-12-07},
	urls = {http://arxiv.org/abs/1310.7968v1,http://arxiv.org/pdf/1310.7968v1},
	collections = {combinatorics,fun-maths-facts,games-to-play-with-friends,the-groups-group},
	url = {http://arxiv.org/abs/1310.7968v1 http://arxiv.org/pdf/1310.7968v1},
	year = 2013,
	urldate = {2020-05-02},
	archivePrefix = {arXiv},
	eprint = {1310.7968},
	primaryClass = {math.GT}
}