arXiv subjects

Edgar A. Bering IV

Publications and source records attributed to Edgar A. Bering IV.

15 recordsLinked to original sources

arXiv2026

Ascending chains of free subgroups in closed hyperbolic and graph 3-manifold groups

Takahasi and Higman independently proved that any ascending chain of subgroups of constant rank in a free group must stabilize. Kapovich and Myasnikov gave a proof of this fact in the language of graphs and Stallings folds. Using profinite techniques, Shusterman extended this constant-rank ascending chain condition to limit groups, which include closed surface groups. Motivated by Kapovich and Myasnikov's proof we provide two new proofs of this ascending chain condition for closed surface groups, and establish the ascending chain condition for free subgroups of constant rank in fundamental groups of closed hyperbolic and graph 3-manifolds. These results are now subsumed by the more general framework established in joint work with Heikamp, Kohav, and Munro which both corrected a mistake in a previous version of this paper and generalized it. The proof for closed hyperbolic 3-manifolds and graph manifolds is preserved in this unpublished note for its direct, geometric approach, which remains valid and particularly transparent.

math.GR↗

arXiv2026

Ascending Chains in 3-Manifold and Relatively Hyperbolic Groups

We prove that any ascending chain of bounded rank subgroups in the fundamental group of a compact $3$-manifold stabilizes. We use geometrization to reduce the proof to fundamental groups of complete, finite-volume hyperbolic $3$-manifolds. To handle this case, we prove the following: In a toral relatively hyperbolic group, any ascending chain of bounded rank, locally relatively quasiconvex subgroups stabilizes. We note this theorem is new even for bounded rank, locally quasiconvex chains in hyperbolic groups.

math.GR↗

arXiv2025

On two-generator subgroups of mapping torus groups

We prove that if $G_ϕ=\langle F, t| t x t^{-1} =ϕ(x), x\in F\rangle$ is the mapping torus group of an injective endomorphism $ϕ: F\to F$ of a free group $F$ (of possibly infinite rank), then every two-generator subgroup $H$ of $G_ϕ$ is either free or a (finitary) sub-mapping torus. As an application we show that if $ϕ\in \mathrm{Out}(F_r)$ (where $r\ge 2$) is a fully irreducible atoroidal automorphism then every two-generator subgroup of $G_ϕ$ is either free or has finite index in $G_ϕ$.

math.GR↗

arXiv2024

Coloring spheres in 3--manifolds

The sphere graph of $M_r$, a connect sum of $r$ copies of $S^1\times S^2$ was introduced by Hatcher as an analog of the curve graph of a surface to study the outer automorphism group of a free group $F_r$. Bestvina, Bromberg, and Fujiwara proved that the chromatic number of the curve graph is finite; bounds were subsequently improved by Gaster, Greene, and Vlamis. Motivated by the analogy, we provide upper and lower bounds for the chromatic number of the sphere graph of $M_r$. As a corollary to the prime decomposition of 3-manifolds, this gives bounds on the chromatic number of the sphere graph for any orientable 3-manifold.

math.GT↗

arXiv2024

An algorithm to decide if an outer automorphism is geometric

An outer automorphism of a free group is geometric if it can be represented by a homeomorphism of a compact surface. Bestvina and Handel gave an algorithmic characterization of geometric irreducible outer automorphisms using relative train tracks in 1995. The general case of detecting geometric outer automorphisms remained open, with a few partial results appearing subsequently. In this paper we give a complete resolution to the problem: an algorithm that can decide if a general outer automorphism is geometric. The algorithm is constructive and produces a realizing surface homeomorphism if one exists. We make use of advances in train-track theory, in conjunction with the Guirardel core of tree actions and Nielsen-Thurston theory for surfaces.

math.GR↗

arXiv2022

Finite rigid sets in sphere complexes

A subcomplex $X\leq \mathcal{C}$ of a simplicial complex is strongly rigid if every locally injective, simplicial map $X\to\mathcal{C}$ is the restriction of a unique automorphism of $\mathcal{C}$. Aramayona and the second author proved that the curve complex of an orientable surface can be exhausted by finite strongly rigid sets. The Hatcher sphere complex is an analog of the curve complex for isotopy classes of essential spheres in a connect sum of $n$ copies of $S^1\times S^2$. We show that there is an exhaustion of the sphere complex by finite strongly rigid sets for all $n\ge 3$ and that when $n=2$ the sphere complex does not have finite rigid sets.

math.GT↗

arXiv2021

Hall's universal group is a subgroup of the abstract commensurator of a free group

P. Hall constructed a universal countable locally finite group U, determined up to isomorphism by two properties: every finite group C is a subgroup of U, and every embedding of C into U is conjugate in U. Every countable locally finite group is a subgroup of U. We prove that U is a subgroup of the abstract commensurator of a finite-rank nonabelian free group.

math.GR↗

arXiv2021

Topological Models of Abstract Commensurators

The full solenoid over a topological space $X$ is the inverse limit of all finite covers. When $X$ is a compact Hausdorff space admitting a locally path connected universal cover, we relate the pointed homotopy equivalences of the full solenoid to the abstract commensurator of the fundamental group $π_1(X)$. The relationship is an isomorphism when $X$ is an aspherical CW complex. If $X$ is additionally a geodesic metric space and $π_1(X)$ is residually finite, we show that this topological model is compatible with the realization of the abstract commensurator as a subgroup of the quasi-isometry group of $π_1(X)$. This is a general topological analogue of work of Biswas, Nag, Odden, Sullivan, and others on the universal hyperbolic solenoid, the full solenoid over a closed surface of genus at least two.

math.GR↗

arXiv2021

Length Function Compatibility for Group Actions on Real Trees

Let $G$ be a finitely generated group. Given two length functions $\ell$ and $m$ of irreducible $G$ actions on real trees $A$ and $B$, when is the point-wise sum $\ell + m$ again the length function of an irreducible $G$ action on a real tree? Guirardel and Levitt showed that additivity is equivalent to the existence of a common refinement of $A$ and $B$, this equivalence is established using Guirardel's core. Moreover, in this case the sum $\ell + m$ is the length function of the common refinement of $A$ and $B$ given explicitly by the Guirardel core. The core can be difficult to compute in general. Behrstock, Bestvina, and Clay give an algorithm for computing the core for free group actions on simplicial trees. In this article we give a geometric characterization of existence of a common refinement that generalizes the criterion underlying Behrstock, Bestvina, and Clay's algorithm, as well as two equivalent characterizations in terms of the associated translation length functions.

math.GR↗

arXiv2020

Explicit special covers of alternating links

Given a prime, alternating link diagram, we build a special cover of the link complement whose degree is bounded by a factorial function of the crossing number. It follows that a subgroup of the link group of that index embeds into right-angled Artin and Coxeter groups. Corollaries of this result include a quantification of residual finiteness, control of the growth of Betti numbers in covers, and an explicit bound on the rank of a Z-module on which the link group acts faithfully.

math.GT↗

arXiv2018

A Criterion For Kolchin Subgroups Of Out(Fr)

This article provides a decidable criterion for when a subgroup of Out(Fr) generated by two Dehn twists consists entirely of polynomially growing elements, answering an earlier question of the author.

math.GR↗

arXiv2018

Uniform independence for Dehn twist automorphisms of a free group

McCarthy's Theorem for the mapping class group of a closed hyperbolic surface states that for any two mapping classes $σ,τ\in \mathrm{Mod}(S)$ there is some power $N$ such that the group $\langle σ^N,τ^N\rangle$ is either free of rank two or abelian, and gives a geometric criterion for the dichotomy. The analogous statement is false in linear groups, and unresolved for outer automorphisms of a free group. Several analogs are known for exponentially growing outer automorphisms satisfying various technical hypothesis. In this article we prove an analogous statement when $σ$ and $τ$ are linearly growing outer automorphisms of $F_r$, and give a geometric criterion for the dichotomy. Further, Hamidi-Tehrani proved that for Dehn twists in the mapping class group this independence dichotomy is \emph{uniform}: $N=4$ suffices. In a similar style, we obtain an $N$ that depends only on the rank of the free group.

math.GR↗

arXiv2016

The random graph embeds in the curve graph of any infinite genus surface

The random graph is an infinite graph with the universal property that any embedding of $G-v$ extends to an embedding of $G$, for any finite graph. In this paper we show that this graph embeds in the curve graph of a surface $Σ$ if and only if $Σ$ has infinite genus, showing that the curve system on an infinite genus surface is "as complicated as possible".

math.GT↗

arXiv2016

On the complexity of finite subgraphs of the curve graph

We say a graph has property $\mathcal{P}_{g,p}$ when it is an induced subgraph of the curve graph of a surface of genus $g$ with $p$ punctures. Two well-known graph invariants, the chromatic and clique numbers, can provide obstructions to $\mathcal{P}_{g,p}$. We introduce a new invariant of a graph, the 'nested complexity length', which provides a novel obstruction to $\mathcal{P}_{g,p}$. For the curve graph this invariant captures the topological complexity of the surface in graph-theoretic terms; indeed we show that its value is $6g-6+2p$, i.e. twice the size of a maximal multicurve on the surface. As a consequence we show that large half-graphs do not have $\mathcal{P}_{g,p}$, and we deduce quantitatively that almost all finite graphs which pass the chromatic and clique tests do not have $\mathcal{P}_{g,p}$. We also reinterpret our obstruction in terms of the first-order theory of the curve graph, and in terms of RAAG subgroups of the mapping class group (following Kim and Koberda). Finally, we show that large multipartite subgraphs cannot have $\mathcal{P}_{g,p}$. This allows us to compute the upper density of the curve graph, and to conclude that clique size, chromatic number, and nested complexity length are not sufficient to determine $\mathcal{P}_{g,p}$.

math.GT↗

arXiv2011

Quantum Algorithms for Invariants of Triangulated Manifolds

One of the apparent advantages of quantum computers over their classical counterparts is their ability to efficiently contract tensor networks. In this article, we study some implications of this fact in the case of topological tensor networks. The graph underlying these networks is given by the triangulation of a manifold, and the structure of the tensors ensures that the overall tensor is independent of the choice of internal triangulation. This leads to quantum algorithms for additively approximating certain invariants of triangulated manifolds. We discuss the details of this construction in two specific cases. In the first case, we consider triangulated surfaces, where the triangle tensor is defined by the multiplication operator of a finite group; the resulting invariant has a simple closed-form expression involving the dimensions of the irreducible representations of the group and the Euler characteristic of the surface. In the second case, we consider triangulated 3-manifolds, where the tetrahedral tensor is defined by the so-called Fibonacci anyon model; the resulting invariant is the well-known Turaev-Viro invariant of 3-manifolds.

quant-ph↗