ON THE COMBINATORIAL ANABELIAN GEOMETRY OF NODALLY NONDEGENERATE OUTER REPRESENTATIONS

By Yuichiro Hoshi, S. Mochizuki. 2011

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Let Σ be a nonempty set of prime numbers. In the present paper, we continue the study, initiated in a previous paper by the second author, of the combinatorial anabelian geometry of semi-graphs of anabelioids of pro-Σ PSC-type, i.e., roughly speak- ing, semi-graphs of anabelioids associated to pointed stable curves. Our first main result is a partial generalization of one of the main combinatorial anabelian results of this previous paper to the case of nodally nondegenerate outer representations, i.e., roughly speak- ing, a sort of abstract combinatorial group-theoretic generalization of the scheme-theoretic notion of a family of pointed stable curves over the spectrum of a discrete valuation ring. We then apply this result to obtain a generalization, to the case of proper hyperbolic curves, of a certain injectivity result, obtained in another paper by the second author, concerning outer automorphisms of the pro-Σ fundamental group of a configuration space associated to a hyper- bolic curve, as the dimension of this configuration space is lowered from two to one. This injectivity allows one to generalize a certain well-known injectivity theorem of Matsumoto to the case of proper hyperbolic curves.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/e52798136cfc94113ca17e93f9c8b14e53100130

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In Geometry and Topology

  1. A. Jaikin‐Zapirain
  2. G. González‐diez
  3. Christopher Voll
  4. Nir Avni
  5. Alexander Stasinski
  6. Avraham Aizenbud

In Mathematical Physics

  1. A. Jaikin‐Zapirain
  2. G. González‐diez
  3. Christopher Voll
  4. Nir Avni
  5. Alexander Stasinski
  6. Avraham Aizenbud

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