arXiv subjects

S. Allen Broughton

Publications and source records attributed to S. Allen Broughton.

5 recordsLinked to original sources

arXiv2026

Explicit Homology Representation for Finite Groups Acting on Riemann Surfaces

Given a finite group $G$ acting orientably on a surface $S$ of genus $\sigma \geq 2$, the group $G$ acts faithfully on the homology group $H_{1}(S;\mathbb{Z})$, preserving the symplectic intersection form. The action on $S$ and the homology is determined by a \emph{generating vector}, a tuple of elements of $G$, generating $G$ and satisfying certain properties. In this note we show how to compute the homology representation, using the generating vector, when $S/G$ has genus 0 and the genus is suitably low. A $2\sigma \times 2\sigma$ representing matrix can be determined for any element in the group, usually for a small set of generators. The matrices are computed with respect to an auto-generated basis for the cellular homology of $S$, using a regular $CW$ structure on $S$, derived from the $G$ action. We demonstrate the application of these results by computing \emph{invariant theta characteristics} of the Riemann surfaces $S$ with the algorithm implemented using Sage.

math.AG↗

arXiv2025

Modular companions in planar one-dimensional equisymmetric strata

Consider, in the moduli space of Riemann surfaces of a fixed genus, the subset of surfaces with non-trivial automorphisms. Of special interest are the numerous subsets of surfaces admitting an action of a given finite group, $G$, acting with a specific signature. In a previous study we declared two Riemann surfaces to be \emph{modular companions} if they have topologically equivalent $G$ actions, and that their $G$ quotients are conformally equivalent orbifolds. In this article we present a geometrically-inspired measure to decide whether two modular companions are conformally equivalent (or how different), respecting the $G$ action. Along the way, we construct a moduli space for surfaces with the specified $G$ action and associated equivariant tilings on these surfaces. We specifically apply the ideas to planar, finite group actions whose quotient orbifold is a sphere with four cone points.

math.GT↗

arXiv2010

Ellipses in translation surfaces

We characterize subgroups of the mapping class group that stabilize a Teichmueller disk in terms of ellipses and strips that are immersed in the associated translation surface. In particular, we show that the space of immersed ellipses/strips that meet at least three cone points is naturally a (non-manifold) 2-dimensional cell complex. The topology of this complex and the geometry of its 0-cells determine the translation surface and its affine diffeomorphism group (up to the kernel of the differential).

math.GT↗

arXiv2010

Cyclic $n$-gonal Surfaces

A cyclic $n$-gonal surface is a compact Riemann surface $X$ of genus $g\geq 2$ admitting a cyclic group of conformal automorphisms $C$ of order $n$ such that the quotient space $X/C$ has genus 0. In this paper, we provide an overview of ongoing research into automorphism groups of cyclic $n$-gonal surfaces. Much of the paper is expository or will appear in forthcoming papers, so proofs are usually omitted. Numerous explicit examples are presented illustrating the computational methods currently being used to study these surfaces.

math.AG↗

arXiv2006

Finite Abelian Subgroups of the Mapping Class Group

The problem of enumeration of conjugacy classes of finite abelian subgroups of the mapping class group $\mathcal{M}_σ$ of a compact closed surface $X$ of genus $σ$ is considered. A complete method of enumeration is achieved for finite elementary abelian subgroups and steps are taken toward enumeration of finite abelian subgroups.

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