On large groups of symmetries of finite graphs embedded in spheres

2017/06/16 by Zimmermann, Bruno P.
#05C10 #57S17 #57S25 #Combinatorics (math.CO) #FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.1706.05187

Abstract

Let G be a finite group acting orthogonally on a pair (S^d,Γ) where Γis a finite, connected graph of genus g>1 embedded in the sphere S^d. The 3-dimensional case d=3 has recently been considered in a paper by C. Wang, S. Wang, Y. Zhang and the present author where for each genus g>1 the maximum order of a G-action on a pair (S³,Γ) is determined and the corresponding graphs Γare classified. In the present paper we consider arbitrary dimensions d and prove that the order of G is bounded above by a polynomial of degree d/2 in g if d is even, and of degree (d+1)/2 if d is odd; moreover the degree d/2 is best possible in even dimensions d. We discuss also the problem, given a finite graph Γand its finite symmetry group, to find the minimal dimension of a sphere into which Γembeds equivariantly as above.

Related

• Graphs in the 3--sphere with maximum symmetry
• Bordered surfaces in the 3-sphere with maximum symmetry
• Extending finite group actions on surfaces over S³
• Embedding surfaces into S³ with maximum symmetry
• The sphere complex of a locally finite graph