H0w have group theory and fractal geometry been combined?

soft-question group-theory fractals geometric-group-theory

Has there been a significant tie made between group theory and fractal geometry? What are some ways that they have been tied together? I've been inspired to ask this question by this image of a free group.

enter image description here


There are a lot of connections in geometric group theory, which studies things like free groups. Every hyperbolic group (an infinite group with a special condition) has a space at infinity that is either a sphere or a fractal. For instance, the free group has a cantor set at infinity. Other groups have Sierpinski curves and Menger sponges.

Space-filling curves also arise in hyperbolic geometry (see the Cannon-Thurston map).

Finally, most fractals can be generated using the ring structure of $\mathbb{C}$.


See the book Indra's pearls by Mumford, Series and Wright for the wonders of Kleinian Groups.

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