The minimal dimension of a sphere with an equivariant embedding of the bouquet of \(g\) circles is \(2g-1\) (Q2136847)

A finite graph \(\Gamma\) is \textit{hyperbolic} if it is connected, of genus \(g \ge 2\) (the rank of its free fundamental group) and without free edges (edges with one vertex of valence 1). The basic problem underlying the present paper is then the following. Given a finite hyperbolic graph \(\Gamma\) with symmetry or automorphism group \(G\), determine the minimal dimension of a sphere \(S^n\) with an equivariant embedding of \(\Gamma\), i.e. an embedding invariant under an orthogonal action of \(G\) on \(S^n\) (in particular, \(G\) must have a faithful \((n+1)\)-dimensional real linear representation). Let \(B_g\) be the finite graph which is a bouquet of \(g\) circles (one vertex and \(g\) closed edges), with symmetry group \((\mathbb Z_2)^g \rtimes S_g\) of order \(2^g g!\); by a result of \textit{S. Wang} and \textit{B. Zimmermann} [Math. Z. 216, No. 1, 83--87 (1994; Zbl 0795.20013)], \(B_g\) is the graph with the largest symmetry group among all finite hyperbolic graphs of genus \(g\), for each \(g\ge 3\). Answering a question of \textit{B. P. Zimmermann} [J. Knot Theory Ramifications 27, No. 3, Article ID 1840011, 8 p. (2018; Zbl 1393.57020)], the main result of the present paper states that the minimal dimension of a sphere with an equivariant embedding of \(B_g\) is \(2g-1\). (On the other hand, the minimal dimension of a faithful action of \((\mathbb Z_2)^g \rtimes S_g\) on a sphere is \(g-1\); also, it is easy to construct an equivariant embedding of \(B_g\) into \(\mathbb R^{2g} = \mathbb R^2 \times \cdots \times \mathbb R^2\) and hence into its 1-point compactification \(S^{2g}\); by the main result of the present paper, the minimal dimension is \(2g-1\).) Concerning dimension 3, finite hyperbolic graphs \(\Gamma\) with large symmetry group and an equivariant embedding into the 3-sphere are classified by \textit{C. Wang} et al. [Discrete Comput. Geom. 59, No. 2, 331--362 (2018; Zbl 1422.57048)]. Finally we note that the analogous problem is interesting also for equivariant embeddings into spheres of compact Riemann or hyperbolic surfaces and their automorphism groups (e.g., what is the minimal dimension of an equivariant embedding of Klein's quartic? (The unique Hurwitz surface of genus 3)).