arXiv subjects
Publications and source records attributed to Elijah Gunther.
A group $G$ splits over a subgroup $C$ if $G$ is either a free product with amalgamation $A \underset{C}{\ast} B$ or an HNN-extension $G=A \underset{C}{\ast} (t)$. We invoke Bass-Serre theory and classify all infinite groups which admit cubic Cayley graphs of connectivity two in terms of splittings over a subgroup.
We begin an investigation of the action of pseudo-Anosov elements of $\mathrm{Out}(\mathbf{F}_{2})$ on the Markoff-type varieties \[ \mathbb{X}_κ:\:x^{2}+y^{2}+z^{2}=xyz+2+κ\] over finite fields $\mathbb{F}_{p}$ with $p$ prime. We first make a precise conjecture about the permutation group generated by $\mathrm{Out}(\mathbf{F}_{2})$ on $\mathbb{X}_{-2}(\mathbb{F}_{p})$ that shows there is no obstruction at the level of the permutation group to a pseudo-Anosov acting `generically'. We prove that this conjecture is sharp. We show that for a fixed pseudo-Anosov $g\in\mathrm{Out}(\mathbf{F}_{2})$, there is always an orbit of $g$ of length $\geq C\log p+O(1)$ on $\mathbb{X}_κ(\mathbb{F}_{p})$ where $C>0$ is given in terms of the eigenvalues of $g$ viewed as an element of $\mathrm{GL}_{2}(\mathbf{Z})$. This improves on a result of Silverman (2007) that applies to general morphisms of quasi-projective varieties. We have discovered that the asymptotic $(p\to\infty)$ behavior of the longest orbit of a fixed pseudo-Anosov $g$ acting on $\mathbb{X}_{-2}(\mathbb{F}_{p})$ is dictated by a dichotomy that we describe both in combinatorial terms and in algebraic terms related to Gauss's ambiguous binary quadratic forms, following Sarnak. This dichotomy is illustrated with numerics, based on which we formulate a precise conjecture.
Doran, Jensen and Giansiracusa showed a bijection between homogeneous elements in the Cox ring of $\overline{M}_{0,n}$ not divisible by any exceptional divisor section, and weighted pure-dimensional simplicial complexes satisfying a zero-tension condition. Motivated by the study of the monoid of effective divisors, the pseudoeffective cone and the Cox ring of $\overline{M}_{0,n}$, we point out a simplification of the zero-tension condition and study the space of balanced complexes. We give examples of irreducible elements in the monoid of effective divisors of $\overline{M}_{0,n}$ for large $n$. In the case of $\overline{M}_{0,7}$, we classify all such irreducible elements arising from nonsingular complexes and give an example of how irreducibility can be shown in the singular case.