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…
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48 results for “free factor system”
Hyperbolicity proven for a specific type of group extension.
problem Proving hyperbolicity of a specific group extension.
method Analyzing ascending HNN extension of groups with a free factor system and an injective endomorphism.
result Ascending HNN extension of a group is hyperbolic relative to a collection of maximal parabolic subgroups.
Relative Currentsmath.GT
In this paper we define currents relative to a free factor system. We prove that a fully irreducible outer automorphism relative to a free factor system acts with uniform north-south dynamics on a subspace of the space of projective relative currents.
We study very small trees from the point of view of reducing systems of free factors, which are analogues of reducing systems of curves for a surface lamination; a non-trivial, proper free factor $F \leq \FN$ reduces if and only if acts on some subtree of with dense orbits. We characterize those trees, call…
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…
The paper explores the geometry and dynamics of free splitting and free factor complexes for groups.
problem Understanding the large scale geometry and dynamics of free splitting and free factor complexes.
method Analyzing the actions of the relative outer automorphism group on these complexes and using tools like the Two Over All Theorem and filling paths.
result Hyperbolicity of the relative free splitting complex and relative free factor complex was proven.
Proves a connectivity conjecture for free groups, showing homotopy type of spheres.
problem Establishing a connectivity conjecture for free groups.
method Provided homotopy-equivalent models of the common basis complex using free factors and sphere systems.
result The common basis complex of a free group of rank n has the homotopy type of a wedge of spheres of dimension 2n-3.
The paper presents a model-free method for stabilizing unknown control systems.
problem Stabilizing unknown control systems in engineering.
method Solving discounted LQR problems with increasing discount factors.
result The method efficiently recovers a stabilizing controller for linear and smooth nonlinear systems.
Measure equivalence rigidity of math.GR
Free group automorphisms group rigidity proven.
problem Proving rigidity of Out(F_N).
method Measure equivalence rigidity, new canonical splittings.
result Superrigidity of Out(F_N).
This is the second in a series of four papers (with research announcement posted on this arXiv) that together develop a decomposition theory for subgroups of Out(F_n). In this paper we relativize the "Kolchin-type theorem" from the work of Bestvina, Feighn, and Handel on the Tits alternative, which describes a decompos…
New space of currents defined for nonabelian free groups and malnormal subgroups.
problem Understanding growth under iteration of outer automorphisms of nonabelian free groups.
method Introducing a new topological space of currents relative to a malnormal subgroup system.
result Currents associated with elements not in conjugates of are dense in the space of currents relative to .
Effective rank rigidity proved for cubulated groups with factor systems.
problem Rank rigidity in cubulated groups with factor systems.
method Exhibiting special pairs of hyperplanes and curtains for skewering.
result Effective form of rank rigidity proved for cubulated groups.
Given a free factor A of the rank n free group F_n, we characterize when the subgroup of Out(F_n) that stabilizes the conjugacy class of A is distorted in Out(F_n). We also prove that the image of the natural embedding of Aut(F_{n-1}) in Aut(F_n) is nondistorted, that the stabilizer in Out(F_n) of the conjugacy class o…
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…
Corrects a 1998 proof about free factors of free groups.
problem Verifying the proof of free factors in free groups.
method Analyzes the geometric realization of free factors.
result Geometric realization is homotopy equivalent to a wedge of spheres.
Connectivity proven in large rank Gromov boundary of free factor complex.
problem Connectivity of Gromov boundary in large rank free factor complex.
method Analyzing Gromov boundary and free factor complex properties.
result Gromov boundary is path connected and locally path connected in large rank.
The free factor complex of rank 4+ fails a combinatorial isoperimetric inequality.
problem Failure of combinatorial isoperimetric inequality in the free factor complex.
method Construction of a coarsely Lipschitz function from the upward link of a free factor to integers.
result A loop in the free factor complex requires linearly growing number of 2-simplices to fill.
The Duffing oscillator's parameters are identified online using variational message passing.
problem Estimating parameters of a nonlinear Duffing oscillator in real-time.
method Variational message passing on a factor graph of the Duffing oscillator's generative model.
result The online inference procedure performs as well as offline methods.
Maps between automorphism groups are isomorphisms for free factor complexes.
problem Understanding the structure of automorphism groups of free factor complexes.
method Establishing isomorphisms between automorphism groups and automorphism groups of free factor complexes.
result Natural maps from to the automorphism group of the free-factor complex are isomorphisms.
We consider the problem of joint modelling of metabolic signals and gene expression in systems biology applications. We propose an approach based on input-output factorial hidden Markov models and propose a structured variational inference approach to infer the structure and states of the model. We start from the class…
The free factor graph for Aut(F_N) is not hyperbolic.
problem Characterizing the geometry of the free factor graph for Aut(F_N).
method Analyzing the quasi-isometric embedding of orbits in the graph of free factors.
result The free factor graph for Aut(F_N) is not hyperbolic.
Conditions for hyperbolic and relatively hyperbolic extensions of free groups using automorphisms with fixed points.
problem Conditions for hyperbolic and relatively hyperbolic extensions of free groups.
method Using dynamics of outer automorphisms on the complex of free factors and investigating the geometry of the extension group.
result 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 from the Handel-Mosher proof of hyperbolicity of the free splitting complex of , thus obtaining an alternative proof of a theorem of Bestvina-Feighn. We also show that under the natural map from the free splitting complex to free factor co…
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.
Study shows top homology group isn't dualizing module for automorphism group of free groups.
problem Identifying the dualizing module for automorphism group of free groups.
method Analyzing the top homology group of the free factor complex.
result The top homology group is not the dualizing module for , especially for .
We provide an effective algorithm for determining whether an element of the outer automorphism group of a free group is fully irreducible. Our method produces a finite list which can be checked for periodic proper free factors.
Study connects group invariants through outer automorphisms and polynomial relations.
problem Understanding polynomial invariants of free-by-cyclic groups.
method Introducing orientable fully irreducible outer automorphisms to relate McMullen polynomial and Alexander polynomial.
result Characterization of when homological stretch factor equals geometric stretch factor.
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 …
Enhanced diffusion sampling improves rare event sampling in biomolecular simulations.
problem Efficiently sampling rare transition events in biomolecular systems.
method Quantitative steering protocols to generate biased ensembles and exact reweighting.
result Fast, accurate, and scalable estimation of equilibrium properties.
Enhanced diffusion sampling tackles rare event sampling in biomolecular simulations.
problem Efficiently sampling rare transition events in biomolecular simulations.
method Quantitative steering protocols to generate biased ensembles, followed by exact reweighting.
result Fast, accurate, and scalable estimation of equilibrium properties for folding free energies.
Most neural networks are trained using first-order optimization methods, which are sensitive to the parameterization of the model. Natural gradient descent is invariant to smooth reparameterizations because it is defined in a coordinate-free way, but tractable approximations are typically defined in terms of coordinate…
Efficient CF approach using fast adaptive PCA for recommender systems.
problem Matrix completion problem in recommender systems.
method Fast adaptive randomized singular value decomposition (SVD) and termination mechanism for latent factors.
result The approach achieves near optimal prediction accuracy with high runtime efficiency.
Random walks on free groups reveal asymmetric expansion factors.
problem Understanding expansion factors in free groups.
method Random walks and BGIP on metric spaces.
result Generic outer automorphisms have different forward and backward expansion factors.
The article constructs a forward utility for markets with multiple default risks.
problem Characterizing forward performance processes in a market with multiple default risks.
method Using Jacod-Pham decomposition and recursive BSDEs, the article constructs a forward utility and proves its existence and uniqueness.
result The article identifies the risk-sensitive long-run growth rate of the optimal wealth process in a stochastic factor model with ergodic dynamics.
We extend some results of [BF12] on subfactor projections to show that the projection of a free factor B to the free factor complex of the free factor A is well-defined with uniformly bound diameter, unless either A is contained in B or A and B are vertex stabilizers of a single splitting of F_n, i.e. they are disjoint…
Auto-sklearn 2.0 simplifies AutoML with meta-learning and meta-feature-free techniques.
problem Designing efficient machine learning pipelines for large datasets under time constraints.
method PoSH Auto-sklearn uses meta-learning and bandit strategy for budget allocation.
result Reduces relative error by up to a factor of 4.5 and improves performance in 10 minutes.
We give an algorithm to compute stable commutator length in free products of cyclic groups which is polynomial time in the length of the input, the number of factors, and the orders of the finite factors. We also describe some experimental and theoretical applications of this algorithm.
Let be a system of free factors of . The group of relative automorphisms is the group given by the automorphisms of that restricted to each are conjugations by elements in . The group of relative outer automorphisms is defined as $Out(F_n;A_1,...,A_k) = Aut(F_n…
By using a notion of a geometric Dehn twist in , we prove that when projections of two -splittings to the free factor complex are far enough from each other in the free factor complex, Dehn twist automorphisms corresponding to the -splittings generate a free group of ra…
We build a collaborative filtering recommender system to restore images with impulse noise for which the noisy pixels have been previously identified. We define this recommender system in terms of a new color image representation using three matrices that depend on the noise-free pixels of the image to restore, and two…
Projections to a graph have bounded diameter for certain group structures.
problem Analyzing the diameter of projections to a specific graph for group structures.
method Analysis of the boundary of hyperbolic groups relative to subgroups and cyclic subgroups.
result The diameter of projections is bounded and depends only on the length bound.
Paper proposes GSSNMF for legal document classification and topic modeling.
problem Lack of methods that can both classify and model topics with guidance.
method Guided Semi-Supervised Non-negative Matrix Factorization (GSSNMF).
result Improves both classification accuracy and topic coherence.
We give a description of the boundary of a complex of free factors that is analogous to E. Klarreich's description of the boundary of a curve complex. The argument uses the geometry of folding paths developed by Bestvina and Feighn as well as structural results about very small trees developed by Coulbois, Hilion, Lust…
Elimination of systemic risk in financial networks by means of a systemic risk transaction taxq-fin.RM
Financial markets are exposed to systemic risk (SR), the risk that a major fraction of the system ceases to function, and collapses. It has recently become possible to quantify SR in terms of underlying financial networks where nodes represent financial institutions, and links capture the size and maturity of assets (l…
SINGD improves KFAC for memory-efficiency and stability in low-precision training.
problem Memory inefficiency and numerical instability of KFAC in low-precision training.
method Formulated inverse-free KFAC update and imposed structures in Kronecker factors.
result SINGD is memory-efficient and numerically robust, often outperforming AdamW in half precision.
New result on critical points of Bethe free energy under deformation retracts.
problem Characterizing critical points of Bethe free energy for complex graphs.
method Analyzing homotopy types and deformation retracts of factor graphs.
result Critical points of Bethe free energy are invariant under deformation retracts.
StockAgent uses AI to simulate real-world stock trading, analyzing external factors and profitability.
problem Investors need to understand how external factors affect stock trading.
method Developed StockAgent, a multi-agent system driven by large language models.
result Identified how external factors impact trading behavior and profitability.
Scl in free productsmath.GT
We study stable commutator length (scl) in free products via surface maps into a wedge of spaces. We prove that scl is piecewise rational linear if it vanishes on each factor of the free product, generalizing the main result in Danny Calegari's paper "Scl, sails and surgery". We further prove that the property of isome…