Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2111.14692

Cones and ping-pong in three dimensions

Gabriel Frieden·Félix Gélinas·Étienne Soucy
View original source ↗Read source document ↗Export metadata ↓

Abstract

We study the hypergeometric group in ${\rm GL}_3(\mathbb{C})$ with parameters $\alpha = (\frac{1}{4}, \frac{1}{2}, \frac{3}{4})$ and $\beta = (0,0,0)$. We give a new proof that this group is isomorphic to the free product $\mathbb{Z}/4\mathbb{Z} * \mathbb{Z}/2\mathbb{Z}$ by exhibiting a ping-pong table. Our table is determined by a simplicial cone in $\mathbb{R}^3$, and we prove that this is the unique simplicial cone (up to sign) for which our construction produces a valid ping-pong table.

Explore related subjects

math.GR

Keep this discovery

BibTeXRIS

Gabriel Frieden, Félix Gélinas, Étienne Soucy. 2021-11-29. Cones and ping-pong in three dimensions. https://doi.org/10.2140/involve.2024.17.11

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Reversibility and its asymptotic counting in Picard group

We investigate reversible elements in the Picard modular group $\mathrm{PSL}(2,\mathbb{Z}[i])$. We show that reversibility coincides with strong reversibility for Kleinian groups, in particular for the Picard group. We classify reversible elements in the Picard group and characterize loxodromic reversible elements up to conjugacy. We prove that each such conjugacy class contains exactly eight special representatives. We also obtain asymptotic estimates for the number of reversible conjugacy classes with bounded trace.

math.GR↗
arXiv2026

Conjugator length in finitely generated groups

We describe all functions $\mathbb{N}\rightarrow \mathbb{N}$ that can be realized, up to the standard equivalence, as conjugator length functions of finitely generated groups. Furthermore, we show that any two increasing functions $f,g\colon \mathbb N\to \mathbb N$ can be simultaneously realized as conjugator length functions of finitely generated, commensurable (in particular, quasi-isometric) groups.

math.GR↗
arXiv2026

The spectrum of conjugator length functions

A recent program tries to find which functions appear as conjugator length functions. In this note, we show that any (computable) increasing function larger than $n$ appears as $\mathrm{Cl}_G$ for some finitely generated (recursively presented) group. On the other hand, we demonstrate that either $\mathrm{Cl}_G$ must be constant or $\mathrm{Cl}_G(n)\succ n$. Combining these, we obtain a complete description of which functions appear as conjugator length functions of finitely generated groups.

math.GR↗