arXiv subjects

Justin Vast

Publications and source records attributed to Justin Vast.

3 recordsLinked to original sources

arXiv2026

Ascending chains of irreducible lattices, bi-reversible automata and affine arithmetic groups

For each $n \geq 2$, we construct an ascending chain of irreducible lattices in the product of $n$ homogeneous trees. Moreover, for each pair of integers $m_1, m_2 \geq 1$, we define explicitly a bi-reversible automaton $\mathcal B$ such that the group $G_{\mathcal B}$ defined by the automaton $\mathcal B$ has finiteness length $m_1$ (i.e. it is of type $\mathrm{F}_{m_1}$ but not of type $\mathrm{FP}_{m_1+1}$), and the group $G_{\mathcal B^*}$ defined by the dual automaton has finiteness length $m_2$. Both constructions rely on the consideration of $S$-arithmetic groups in the affine group of a global function field.

math.GR↗

arXiv2026

Fractal anti-tori

Let $\Gamma$ be a group acting properly and cocompactly on the product of two trees $T_1$ and $T_2$. An anti-torus is a non-periodic flat plane in $T_1 \times T_2$ that is the convex hull of two secant periodic lines. That notion was introduced by Dani Wise as a tool to show that $\Gamma$ is irreducible. We establish a new criterion ensuring the existence of anti-tori, and use it to prove that if $\Gamma$ is an $S$-arithmetic lattice in a product of simple algebraic groups of rank one, then $T_1\times T_2$ contains anti-tori. As a byproduct, we obtain a sufficient condition ensuring that a group defined by a bi-reversible automaton contains non-abelian free sub-semigroups. We also introduce a new class of irreducible lattices acting regularly on the vertex set of a product of two trees, and containing anti-tori that are fractal aperiodic tilings of the plane. This establishes a connection between lattices in products of trees and substitution tilings.

math.GR↗

arXiv2026

On the self-similarity of rational power series with matrix coefficients

Let $p$ be a prime, let $d \geq 1$ be an integer and $A$ be the algebra of square matrices of size $d$ over the field of order $p$. Let $P, Q \in A[x_1, \dots x_n]$ be polynomials in $n$ indeterminates with coefficients in $A$, such that $Q$ is invertible in $ A[\![x_1, \dots, x_n]\!]$. Let also $\mathcal M \colon \mathbf Z^n \to A$ be the map associating to the $n$-tuple of integers $(\alpha_1, \dots, \alpha_n)$ the coefficient of the monomial $x_1^{\alpha_1} \dots x_n^{\alpha_n}$ in the development of the rational fraction $PQ^{-1}$ as a power series (the support of $\mathcal M$ is contained in $\mathbf N^n$). Our main result ensures that the map $\mathcal M$, viewed as a tiling of $\mathbf R^n$ by unit cubes with color set $A$, is self-similar. The self-similarity is expressed in terms of invariance under substitutions. By specializing to $d=1$, $n=2$, $P=1$ and $Q =1-x_1-x_2$, we recover the well-known self-similarity feature of the binomial coefficients modulo $p$.

math.CO↗