Some questions about the dimension of a group action

This article discusses three families of groups: Z Zⁿ, PL(Iⁿ),and PL(Sⁿ) (the last two being the families of groups of piecewise-linearhomeomorphisms of standard n-dimensional spaces). It is shownthat for positive n N, Z Zⁿ embeds in PL(Iⁿ), which embedsin PL(Sⁿ). It is known that Z Z² fails to embed in PL(I¹),and this article extends that previous result to show that Z Z² also fails to embed in PL(S¹). The nature of the proofsof these embedding and non-embedding results hints that theremay be corresponding non-embedding results in higher dimensions.     

A family of non-isomorphism results

We calculate the local groups of germs associated with the higher dimensional R. Thompson groups nV. For a given \({n\in N\cup\left\{\omega\right\}}\) , these groups of germs are free abelian groups of rank r, for r ≤ n (there are some groups of germs associated with nV with rank precisely k for each index 1 ≤ k ≤ n). By Rubin’s theorem, any conjectured isomorphism between higher dimensional R. Thompson groups induces an isomorphism between associated groups of germs. Thus, if m ≠ n the groups mV and nV cannot be isomorphic. This answers a question of Brin.     

Some isomorphism results for Thompson-like groups <Emphasis Type="Italic">V</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>(<Emphasis Type="Italic">G</Emphasis>)

The natural automorphism group of a translation surface is its group of translations. For finite translation surfaces of genus g ≥ 2 the order of this group is naturally bounded in terms of g due to a Riemann–Hurwitz formula argument. In analogy with classical Hurwitz surfaces, we call surfaces which achieve the maximal bound Hurwitz translation surfaces. We study for which g there exist Hurwitz translation surfaces of genus g.