arXiv subjects

Grace S. Garden

Publications and source records attributed to Grace S. Garden.

3 recordsLinked to original sources

arXiv2024

An invitation to Culler-Shalen theory in arbitrary characteristic

In the seminal work of Culler and Shalen from 1983, essential surfaces in 3-manifolds are associated to ideal points of their $\text{SL}_2(\mathbb{C})$-character varieties, and connections between the algebraic geometry of the character variety and the topology of the 3-manifold are established via group actions on trees. Here, we lay a general foundation for this theory in arbitrary characteristic by using the same approach instead over an arbitrary algebraically closed field. Examples include a change in the $A$-polynomial in characteristic 2, a closed Haken hyperbolic 3-manifold with no detected essential surface, and a closed Haken hyperbolic 3-manifold with an essential surface only detected in characteristic 2.

math.GT↗

arXiv2024

Essential tori in 3-manifolds not detected in any characteristic

Infinite families of 3-dimensional closed graph manifolds and closed Seifert fibered spaces are exhibited, each member of which contains an essential torus not detected by ideal points of the variety of $\text{SL}_2(\mathbb{F})$-characters over any algebraically closed field $\mathbb{F}$.

math.GT↗

arXiv2023

Earthquakes on the once-punctured torus

We study earthquake deformations on Teichmüller space associated with simple closed curves of the once-punctured torus. We describe two methods to get an explicit form of the earthquake deformation for any simple closed curve. The first method is rooted in linear recurrence relations, the second in hyperbolic geometry. The two methods align, providing both an algebraic and geometric interpretation of the earthquake deformations. We convert the expressions to other coordinate systems for Teichmüller space to examine earthquake deformations further. Two families of curves are used as examples. Examining the limiting behaviour of each gives insight into earthquakes about measured geodesic laminations, of which simple closed curves are a special case.

math.GT↗