# GROUP-THEORETICITY OF NUMERICAL INVARIANTS AND DISTINGUISHED SUBGROUPS OF CONFIGURATION SPACE GROUPS

@inproceedings{Hoshi2021GROUPTHEORETICITYON, title={GROUP-THEORETICITY OF NUMERICAL INVARIANTS AND DISTINGUISHED SUBGROUPS OF CONFIGURATION SPACE GROUPS}, author={Yuichiro Hoshi and Arata Minamide and Shinichi Mochizuki}, year={2021} }

Let Σ be a set of prime numbers which is either of cardinality one or equal to the set of all prime numbers. In this paper, we prove that various objects that arise from the geometry of the configuration space of a hyperbolic curve over an algebraically closed field of characteristic zero may be reconstructed group-theoretically from the pro-Σ fundamental group of the configuration space. Let X be a hyperbolic curve of type (g, r) over a field k of characteristic zero. Thus, X is obtained by…

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## References

SHOWING 1-10 OF 32 REFERENCES

On the combinatorial cuspidalization of hyperbolic curves

- Mathematics
- 2010

In this paper, we continue our study of the pro-Σ fundamental groups of configuration spaces associated to a hyperbolic curve, where Σ is either the set of all prime numbers or a set consisting of a…

Topics surrounding the combinatorial anabelian geometry of hyperbolic curves I: Inertia groups and profinite Dehn twists

- Mathematics
- 2012

Let Σ be a subset of the set of prime numbers which is either equal to the entire set of prime numbers or of cardinality one. In the present paper, we continue our study of the pro-Σ fundamental…

Galois rigidity of pure sphere braid groups and profinite calculus

- Mathematics
- 1994

Let C be a class of finite groups closed under the for- mation of subgroups, quotients, and group extensions. For an algebraic variety X over a number field k, let π C 1 (X) denote the (C-modified)…

The Absolute Anabelian Geometry of Canonical Curves

- Mathematics
- 2003

In this paper, we continue our study of the issue of the ex- tent to which a hyperbolic curve over a finite extension of the field of p-adic numbers is determined by the profinite group structure of…

The local pro-p anabelian geometry of curves

- Mathematics
- 1999

Let X be a connected scheme. Then one can associate (after Grothendieck) to X its algebraic fundamental group π1(X). This group π1(X) is a profinite group which is uniquely determined (up to inner…

ON THE COMBINATORIAL ANABELIAN GEOMETRY OF NODALLY NONDEGENERATE OUTER REPRESENTATIONS

- Mathematics
- 2011

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…

THE ALGEBRAIC AND ANABELIAN GEOMETRY OF CONFIGURATION SPACES

- Mathematics
- 2008

In this paper, we study the pro-Σ fundamental groups of configuration spaces, where Σ is either the set of all prime numbers or a set consisting of a single prime number. In particular, we show, via…

TOPICS SURROUNDING THE COMBINATORIAL ANABELIAN GEOMETRY OF HYPERBOLIC CUR VES II: TRIPODS AND COMBINATORIAL CUSPIDALIZATION

- Mathematics
- 2014

Letbe a subset of the set of prime numbers which is either equal to the entire set of prime numbers or of cardinal- ity one. In the present paper, we continue our study of the pro-� fundamental…

Galois rigidity of pro- pure braid groups of algebraic curves

- Mathematics
- 1998

In this paper, Grothendieck’s anabelian conjecture on the pro-l fundamental groups of configuration spaces of hyperbolic curves is reduced to the conjecture on those of single hyperbolic curves. This…