arXiv research

The free factor graph for Aut(F_N) is not hyperbolic.

We show that the Gromov boundary of the free factor graph for the free group Fn with n>2 generators is the space of equivalence classes of minimal very small indecomposable projective Fn-trees without point stabilizer containing a free factor equipped with a quotient topology. Here two such trees are equivalent if the …

We give upper bounds, linear in rank, to the topological dimensions of the Gromov boundaries of the intersection graph, the free factor graph and the cyclic splitting graph of a finitely generated free group.

New complex connects graph separability to group properties.

New hyperbolic graph constructed from projections of free splitting graph.

New result on critical points of Bethe free energy under deformation retracts.

Projections to a graph have bounded diameter for certain group structures.

We characterize strongly Morse quasi-geodesics in Outer space as quasi-geodesics which project to quasi-geodesics in the free factor graph. We define convex cocompact subgroups of $Out(F_n)$ as subgroups such that an orbit map in the free factor graph is a quasi-isometric embedding, and we characterize such groups via …

The study examines the stretch factors of outer automorphisms and their latent symmetry.

Many graph clustering quality functions suffer from a resolution limit, the inability to find small clusters in large graphs. So called resolution-limit-free quality functions do not have this limit. This property was previously introduced for hard clustering, that is, graph partitioning. We investigate the resolution-…

The end compactification |Γ| of the locally finite graph Γis the union of the graph and its ends, endowed with a suitable topology. We show that π_1(|Γ|) embeds into a nonstandard free group with hyperfinitely many generators, i.e. an ultraproduct of finitely generated free groups, and that the embedding we construct f…

Given a countable group $G$ splitting as a free product $G=G_1\ast\dots\ast G_k\ast F_N$, we establish classification results for subgroups of the group $Out(G,\mathcal{F})$ of all outer automorphisms of $G$ that preserve the conjugacy classes of each $G_i$. We show that every finitely generated subgroup $H\subseteq Ou…

For every finite graph $Γ$, we define a simplicial complex associated to the outer automorphism group of the RAAG $A_Γ$. These complexes are defined as coset complexes of parabolic subgroups of $Out^0(A_Γ)$ and interpolate between Tits buildings and free factor complexes. We show that each of these complexes is homotop…

We present a joint message passing approach that combines belief propagation and the mean field approximation. Our analysis is based on the region-based free energy approximation method proposed by Yedidia et al. We show that the message passing fixed-point equations obtained with this combination correspond to station…

By using a notion of a geometric Dehn twist in $\sharp_k(S^2 \times S^1)$, we prove that when projections of two $\mathbb{Z}$-splittings to the free factor complex are far enough from each other in the free factor complex, Dehn twist automorphisms corresponding to the $\mathbb{Z}$-splittings generate a free group of ra…

We introduce the co-surface graph $\mathcal{CS}$ of a finitely generated free group $\mathbb{F}$ and use it to study the geometry of hyperbolic group extensions of $\mathbb{F}$. Among other things, we show that the Gromov boundary of the co-surface graph is equivariantly homeomorphic to the space of free arational $\ma…

3-manifold groups' word problem solved in nearly linear time.

Belief propagation (BP) can do exact inference in loop-free graphs, but its performance could be poor in graphs with loops, and the understanding of its solution is limited. This work gives an interpretable belief propagation rule that is actually minimization of a localized $α$-divergence. We term this algorithm as $α…

We prove that the palindromic width of HNN extension of a group by proper associated subgroups is infinite. We also prove that the palindromic width of the amalgamated free product of two groups via a proper subgroup is infinite (except when the amalgamated subgroup has index two in each of the factors). Combining thes…

We show that the Gromov boundary of the free product of two infinite hyperbolic groups is uniquely determined up to homeomorphism by the homeomorphism types of the boundaries of its factors. We generalize this result to graphs of hyperbolic groups over finite subgroups. Finally, we give a necessary and sufficient condi…

Let $f\colon \mathbb{S}^1\rightarrow G$ be a surjective map from the standard unit circle to a graph $G$ such that the pre-image of each point has diameter less than $\varepsilon$. If $\varepsilon$ is small enough, does $f$ split as a free factor in $π_1(G)$?

New algorithm minimizes FE objectives for synthetic AIF agents.

The pants graph of a free group is constructed and studied.

Corrects a 1998 proof about free factors of free groups.

Connectivity proven in large rank Gromov boundary of free factor complex.

We show that the complex of free factors of a free group of rank n > 1 is homotopy equivalent to a wedge of spheres of dimension n-2. We also prove that for n > 1, the complement of (unreduced) Outer space in the free splitting complex is homotopy equivalent to the complex of free factor systems and moreover is (n-2)-c…

The free factor complex of rank 4+ fails a combinatorial isoperimetric inequality.

Tree-AMP simplifies inference in complex tree-structured models.

Maps between automorphism groups are isomorphisms for free factor complexes.

We propose algorithms for approximate filtering and smoothing in high-dimensional Factorial hidden Markov models. The approximation involves discarding, in a principled way, likelihood factors according to a notion of locality in a factor graph associated with the emission distribution. This allows the exponential-in-d…

Proposes a low-rank bilinear pooling model for link prediction in knowledge graphs.

The paper explores linearly free graphs and their embeddings into 3D space.

The group $\Out$ of outer automorphisms of the free group has been an object of active study for many years, yet its geometry is not well understood. Recently, effort has been focused on finding a hyperbolic complex on which $\Out$ acts, in analogy with the curve complex for the mapping class group. Here, we focus on o…

The study proves conjecture for specific Artin groups.

DAOR efficiently embeds graphs without tuning, improving speed and interpretability.

Given a free group $F_k$ of rank $k\ge 2$ with a fixed set of free generators we associate to any homomorphism $φ$ from $F_k$ to a group $G$ with a left-invariant semi-norm a generic stretching factor, $λ(φ)$, which is a non-commutative generalization of the translation number. We concentrate on the situation when $φ:F…

Conditions for hyperbolic and relatively hyperbolic extensions of free groups using automorphisms with fixed points.

We show how to derive hyperbolicity of the free factor complex of $F_N$ from the Handel-Mosher proof of hyperbolicity of the free splitting complex of $F_N$, thus obtaining an alternative proof of a theorem of Bestvina-Feighn. We also show that under the natural map $τ$ from the free splitting complex to free factor co…

Agent uses message passing to optimize robot navigation, balancing exploration and exploitation.

Graph neural networks speed up nonnegative matrix factorization.

Study abelian factors in Lie algebras from graph edge labels.

An embedding of a graph into $\mathbb{R}^3$ is said to be linear, if any edge of the graph is sent to be a line segment. And we say that an embedding $f$ of a graph $G$ into $\mathbb{R}^3$ is free, if $π_1(\mathbb{R}^3-f(G))$ is a free group. It was known that for any complete graph its linear embedding is always free.…

Paper proves flatness of anisotropic minimal graphs in half-spaces.

We simplify word embeddings by removing sigmoid in SGNS, revealing connections to hyperbolic spaces.

We develop the geometry of folding paths in Outer space and, as an application, prove that the complex of free factors of a free group of finite rank is hyperbolic.

Extends graph factor system to quasi-median graphs.

A wide class of machine learning algorithms can be reduced to variable elimination on factor graphs. While factor graphs provide a unifying notation for these algorithms, they do not provide a compact way to express repeated structure when compared to plate diagrams for directed graphical models. To exploit efficient t…

In this paper we prove that a fully irreducible outer automorphism relative to a non-exceptional free factor system acts loxodromically on the relative free factor complex as defined by Handel and Mosher. We also prove a north-south dynamic result for the action of such outer automorphisms on the closure of relative ou…