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Alex Furman

Publications and source records attributed to Alex Furman.

At least 19 recordsLinked to original sources
arXiv2025

Quotients of Poisson boundaries, entropy, and spectral gap

Poisson boundary is a measurable $\Gamma$-space canonically associated with a group $\Gamma$ and a probability measure $\mu$ on it. The collection of all measurable $\Gamma$-equivariant quotients, known as $\mu$-boundaries, of the Poisson boundary forms a partially ordered set, equipped with a strictly monotonic non-negative function, known as Furstenberg or differential entropy. In this paper we demonstrate the richness and the complexity of this lattice of quotients for the case of free groups and surface groups and rather general measures. In particular, we show that there are continuum many unrelated $\mu$-boundaries at each, sufficiently low, entropy level, and there are continuum many distinct order-theoretic cubes of $\mu$-boundaries. These $\mu$-boundaries are constructed from dense linear representations $\rho:\Gamma\to G$ to semi-simple Lie groups, like $\PSL_2(\bbC)^d$ with absolutely continuous stationary measures on $\hat\bbC^d$.

math.GR↗
arXiv2025

Lyapunov spectrum via boundary theory I

This paper is concerned with the Lyapunov spectrum for measurable cocycles over an ergodic pmp system taking values in semi-simple real Lie groups. We prove simplicity of the Lyapunov spectrum and its continuity under certain perturbations for a class systems that includes many familiar examples. Our framework uses some soft qualitative assumptions, and does not rely on symbolic dynamics. We use ideas from boundary theory that appear in the study of super-rigidity to deduce our results. This gives a new perspective even on the most studied case of random matrix products. The current paper introduces the general framework and contains the proofs of the main results and some basic examples. In a follow up paper we discuss further examples.

math.DS↗
arXiv2021

Quasi-Fuchsian vs Negative curvature metrics on surface groups

We compare two families of left-invariant metrics on a surface group $\Gamma=\pi_1(\Sigma)$ in the context of course-geometry. One family comes from Riemannian metrics of negative curvature on the the surface $\Sigma$, and another from quasi-Fuchsian representations of $\Gamma$. We show that the Teichmuller space $\Teich(\Sigma)$ is the only common part of these two families, even when viewed from the coarse-geometric perspective.

math.GT↗
arXiv2021

Central Limit Theorem for Cocycles over Hyperbolic Systems

We prove a Central Limit Theorem (CLT) in the non-commutative setting of random matrix products where the underlying process is driven by a subshift of finite type (SFT) with Markov measure. We use the martingale method introduced by Y. Benoist and J.F. Quint in the iid setting.

math.PR↗
arXiv2019

Wisdom of collaborators: a peer-review approach to performance appraisal

Individual performance and reputation within a company are major factors that influence wage distribution, promotion and firing. Due to the complexity and collaborative nature of contemporary business processes, the evaluation of individual impact in the majority of organizations is an ambiguous and non-trivial task. Existing performance appraisal approaches are often affected by individuals biased judgements, and organizations are dissatisfied with the results of evaluations. We assert that employees can provide accurate measurement of their peer performance in a complex collaborative environment. We propose a novel metric, the Peer Rank Score (PRS), that evaluates individual reputations and the non-quantifiable individual impact. PRS is based on pairwise comparisons of employees. We show high robustness of the algorithm on simulations and empirically validate it for a genetic testing company on more than one thousand employees using peer reviews over the course of three years.

cs.SI↗
arXiv2018

An extension of Margulis' Super-Rigidity Theorem

We give an extension of Margulis' Super-Rigidity for higher rank lattices. In our approach the target group could be defined over any complete valued field. Our proof is based on the notion of Algebraic Representation of Ergodic Actions.

math.GR↗
arXiv2018

Margulis' Super-Rigidity Theorem for non-lattice

We give a Super-Rigidity theorem a la Margulis which applies for a wider class of groups. In particular it applies to subgroups which are not assumed to be lattices in the ambient group. Our proof is based on the notion of Algebraic Representation of Ergodic Actions.

math.GR↗
arXiv2018

Interpreting Electrical-Resistivity Tomography measurements using Neural Network

Electrical Resistivity Tomography (ERT) has been extensively used for imaging the subsurface resistivity distribution and structure. Over the years, many algorithms have been developed in order to solve the subsurface resistivity distribution from the ERT measurements. In this paper a new method for interpreting the ERT measurements is presented. Using supervised learning to train a neural network, we are able to interpret the ERT measurement into a 2D image of the underground resistivity up to depths of 50 meters while using a simple Wenner-Schlumberger survey of 96 electrodes with 1 meter spacing. The neural network is trained and tested using simulative data and it is shown to have superior results over a well established inversion method.

physics.geo-ph↗
arXiv2018

Super-Rigidity and non-linearity for lattices in products

We prove a super-rigidity result for algebraic representations over complete fields of irreducible lattices in product of groups and lattices with dense commensurator groups. We derive some criteria for non-linearity of such groups.

math.GR↗
arXiv2017

Lattice envelopes

We introduce a class of countable groups by some abstract group-theoretic conditions. It includes linear groups with finite amenable radical and finitely generated residually finite groups with some non-vanishing $\ell^2$-Betti numbers that are not virtually a product of two infinite groups. Further, it includes acylindrically hyperbolic groups. For any group $\Gamma$ in this class we determine the general structure of its possible lattice embeddings, i.e. of all compactly generated, locally compact groups that contain $\Gamma$ as a lattice. This leads to a precise description of possible non-uniform lattice embeddings of groups in this class. Further applications include the determination of possible lattice embeddings of fundamental groups of closed manifolds with pinched negative curvature.

math.GR↗
arXiv2017

Some ergodic properties of metrics on hyperbolic groups

Let $\Gamma$ be a non-elementary Gromov-hyperbolic group, and $\partial \Gamma$ denote its Gromov boundary. We consider $\Gamma$-invariant proper $\delta$-hyperbolic, quasi-convex metric $d$ on $\Gamma$, and the associated Patterson-Sullivan measure class $[\nu]$ on $\partial^{(2)}\Gamma$, and its square $[\nu\times\nu]$ on $\partial^{(2)}\Gamma$ -- the space of distinct pairs of points on the boundary. We construct an analogue of a geodesic flow to study ergodicity properties of the $\Gamma$-actions on $(\partial\Gamma,\nu)$ and on $(\partial^{(2)}\Gamma,[\nu\times\nu])$. We also prove some ergodic theorems for $\Gamma$-actions guided by the geometry of $(\Gamma,d)$.

math.DS↗
arXiv2017

An adelic arithmeticity theorem for lattices in products

We prove that, under mild assumptions, a lattice in a product of semi-simple Lie group and a totally disconnected locally compact group is, in a certain sense, arithmetic. We do not assume the lattice to be finitely generated or the ambient group to be compactly generated.

math.GR↗
arXiv2015

Asymptotic Shapes for Ergodic Families of Metrics on Nilpotent Groups

Let Gamma be a finitely generated nilpotent group. We consider three closely related problems: (i) the asymptotic cone for an equivariant ergodic family of inner metrics on Gamma, generalizing Pansu's theorem; (ii) the limit shapes for First Passage Percolation for general (not necessarily independent) ergodic processes on edges of a Cayley graph of Gamma; (iii) a sub-additive ergodic theorem over a general ergodic Gamma-action. The limiting objects are given in terms of a Carnot-Caratheodory metric on the graded nilpotent group associated to the Mal'cev completion of Gamma.

math.GR↗
arXiv2014

On the structure and arithmeticity of lattice envelopes

We announce results about the structure and arithmeticity of all possible lattice embeddings of a class of countable groups which encompasses all linear groups with simple Zariski closure, all groups with non-vanishing first l2-Betti number, word hyperbolic groups, and, more general, convergence groups.

math.GR↗
arXiv2014

Boundaries, rigidity of representations, and Lyapunov exponents

In this paper we discuss some connections between measurable dynamics and rigidity aspects of group representations and group actions. A new ergodic feature of familiar group boundaries is introduced, and is used to obtain rigidity results for group representations and to prove simplicity of Lyapunov exponents for some dynamical systems.

math.DS↗
arXiv2014

Algebraic Representations of Ergodic Actions and Super-Rigidity

We revisit Margulis-Zimmer Super-Rigidity and provide some generalizations. In particular we obtain super-rigidity results for lattices in higher-rank groups or product of groups, targeting at algebraic groups over arbitrary fields with absolute values. We also obtain cocycle super-rigidity results for a wide class of groups with respect to mixing actions. Our approach is based on a systematic study of algebraic representations of ergodic actions.

math.GR↗