Abstract
We prove that the first homology group of every planar locally finite transitive graph $G$ is finitely generated as an $\Aut(G)$-module and we prove a similar result for the fundamental group of locally finite planar Cayley graphs. Corollaries of these results include Droms's theorem that planar groups are finitely presented and Dunwoody's theorem that planar locally finite transitive graphs are accessible.
Highlights
A finitely generated group is planar if it has some locally finite planar Cayley graph
The fundamental group of G has a generating set consisting of finitely many FS -orbits
Every planar locally finite transitive graph G has a set of cycles that generates the first homology group and consists of finitely many Aut(G)-orbits
Summary
A finitely generated group is planar if it has some locally finite planar Cayley graph. Every planar locally finite transitive graph G has a set of cycles that generates the first homology group and consists of finitely many Aut(G)-orbits. We will prove our first main step in Section 4: we will see that in finitely separable 3-connected planar graphs the space of all closed walks has a nested generating set (Theorem 13).
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