arXiv research

New space of currents defined for nonabelian free groups and malnormal subgroups.

Proves existence of certain subgroups in hyperbolic groups.

The aim of the current paper is to explore the implications on the group $G$ of the non-vanishing of the cohomology in degree one of one of its representation $π$, given some mixing conditions on $π$. In one direction, harmonic cocycles are used to show that the FC-centre should be finite (for mildly mixing unitary rep…

We survey the problem of separation under conjugacy and malnormality of the abelian peripheral subgroups of an orientable, irreducible $3$-manifold $X$. We shall focus on the relation between this problem and the existence of acylindrical splittings of $π_1(X)$ as an amalgamated product or HNN-extension along the abeli…

The intersection pattern of the translates of the limit set of a quasi-convex subgroup of a hyperbolic group can be coded in a natural incidence graph, which suggests connections with the splittings of the ambient group. A similar incidence graph exists for any subgroup of a group. We show that the disconnectedness of …

Complex captures group properties, invariant under quasi-isometry.

A line pattern in a free group $F$ is defined by a malnormal collection of cyclic subgroups. Otal defined a decomposition space $\mathcal{D}$ associated to a line pattern. We provide an algorithm that computes a presentation for the Čech cohomology of $\mathcal{D}$, thought of as a $F$-module. This answers a relative v…

We show that for groups acting acylindrically on simplicial trees the $K$- and $L$-theoretic Farrell-Jones Conjecture relative to the family of subgroups consisting of virtually cyclic subgroups and all subconjugates of vertex stabilisers holds. As an application, for amalgamated free products acting acylindrically on …

Wise's Quasiconvex Hierarchy Theorem classifying hyperbolic virtually compact special groups in terms of quasiconvex hierarchies played an essential role in Agol's proof of the Virtual Haken Conjecture. Answering a question of Wise, we construct a new virtual quasiconvex hierarchy for relatively hyperbolic virtually co…

We give an alternate proof of Wise's Malnormal Special Quotient Theorem (MSQT), avoiding cubical small cancellation theory. We also show how to deduce Wise's Quasiconvex Hierarchy Theorem from the MSQT and theorems of Hsu--Wise and Haglund--Wise.

The paper proves that relative Dehn functions are invariant under quasi-isometry.

New insights into the structure of blown-up corona of hyperbolic groups.

We introduce the notion of controlled Floyd separation between geodesic rays starting at the identity in a finitely generated group G. Two such geodesic rays are said to be Floyd separated with respect to quasigeodesics if the (Floyd) length of c-quasigeodesics (for fixed but arbitrary c) joining points on the geodesic…

For any finitely generated, non-elementary, torsion-free group $G$ that is hyperbolic relative to $\mathbb P$, we show that there exists a group $G^*$ containing $G$ such that $G^*$ is hyperbolic relative to $\mathbb P$ and $G$ is not relatively quasiconvex in $G^*$. This generalizes a result of I. Kapovich for hyperbo…

We study automorphisms of a relatively hyperbolic group G. When G is one-ended, we describe Out(G) using a preferred JSJ tree over subgroups that are virtually cyclic or parabolic. In particular, when G is toral relatively hyperbolic, Out(G) is virtually built out of mapping class groups and subgroups of GL_n(Z) fixing…

Graphically discrete groups have strong rigidity properties.

Answering a question asked by Agol and Wise, we show that a desired stronger form of Wise's malnormal special quotient theorem does not hold. The counterexamples are generalizations of triangle groups, built using the Ramanujan graphs constructed by Lubotzky--Phillips--Sarnak.

Hierarchically hyperbolic spaces (HHSs) are a large class of spaces that provide a unified framework for studying the mapping class group, right-angled Artin and Coxeter groups, and many 3--manifold groups. We investigate strongly quasiconvex subsets in this class and characterize them in terms of their contracting pro…

Given a representation of a link group, we introduce a trilinear form, as a topological invariant. We show that, if the link is either hyperbolic or a knot with malnormality, then the trilinear form equals the pairing of the (twisted) triple cup product and the fundamental relative 3-class. Further, we give some exampl…

The growing capability and accessibility of machine learning has led to its application to many real-world domains and data about people. Despite the benefits algorithmic systems may bring, models can reflect, inject, or exacerbate implicit and explicit societal biases into their outputs, disadvantaging certain demogra…

The paper defines conditions for a free-by-free group to be hyperbolic.

The paper tackles fairness in forecasting and learning linear dynamical systems.

Constructs flows on quotients of Lie groups for Anosov subgroups.

We propose a sliding surface for systems on the Lie group $SO(3)\times \mathbb{R}^3$ . The sliding surface is shown to be a Lie subgroup. The reduced-order dynamics along the sliding subgroup have an almost globally asymptotically stable equilibrium. The sliding surface is used to design a sliding-mode controller for t…

The minimal standardizer of a curve system on a punctured disk is the minimal braid that transforms it into a system formed only by round curves. We give an algorithm to compute it in a geometrical way. Then, we generalize this problem algebraically to parabolic subgroups of Artin-Tits groups of spherical type and we s…

Study growth rates of subgroups in groups with a constricting element.

New method learns from subgroup feedback in complex systems.

The paper solves the conjugacy problem in a specific braid group quotient and finds infinite virtually cyclic subgroups.

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…

A new method improves AI fairness assessment by estimating performance across intersectional subgroups.

Study generalizes non-interaction theorems for relativistic systems.

Hyperbolicity proven for a specific type of group extension.

GAME improves matrix completion by considering subgroup-specific latent structures.

We prove that there exists a positive, explicit function $F(k, E)$ such that, for any group $G$ admitting a $k$-acylindrical splitting and any generating set $S$ of $G$ with $\mathrm{Ent}(G,S)<E$, we have $|S| \leq F(k, E)$. We deduce corresponding finiteness results for classes of groups possessing acylindrical splitt…

Study preserves symplectic structure in forced discrete mechanical systems.

R2P method identifies homogeneous and heterogeneous subgroups for better treatment effect estimation.

Classifies geodesic orbit spaces with abelian isotropy subgroups.

A new metric MSD detects bias in datasets efficiently.

The paper consists of two parts. In the first one we show that a relatively hyperbolic group $G$ splits as a star graph of groups whose central vertex group is finitely generated and the other vertex groups are maximal parabolic subgroups. As a corollary we obtain that every group which admits 3-discontinuous and 2-coc…

Given a system of equations in a "random" finitely generated subgroup of the braid group, we show how to find a small ordered list of elements in the subgroup, which contains a solution to the equations with a significant probability. Moreover, with a significant probability, the solution will be the first in the list.…

Suppose that $(W,S)$ is a Coxeter system with associated Artin group $A$ and with a simplicial complex $L$ as its nerve. We define the notion of a "standard abelian subgroup" in $A$. The poset of such subgroups in $A$ is parameterized by the poset of simplices in a certain subdivision $L_\oslash$ of $L$. This complex o…

Study invariant measures on measured laminations for subgroups of mapping class group.

New holistic approach measures sample-level adversarial vulnerability for trustworthy systems.

We prove a global Birkhoff decomposition for almost split real forms of loop groups, when an underlying finite dimensional Lie group is compact. Among applications, this shows that the dressing action - by the whole subgroup of loops which extend holomorphically to the exterior disc - on the $U$-hierarchy of the ZS-AKN…

In this paper, we investigate the effect of machine learning based anonymization on anomalous subgroup preservation. In particular, we train a binary classifier to discover the most anomalous subgroup in a dataset by maximizing the bias between the group's predicted odds ratio from the model and observed odds ratio fro…

Improves fairness in machine learning by adding underrepresented group data.

The paper introduces a model to measure ASR fairness, addressing key issues.

In this paper, we present the point symmetry group of three-dimensional homogeneous Helmholtz equation, when we consider the cylindrical coordinate system. In continuation, we present a complete set of functionally independent invariants of the equation along with the form of the general solution provided by these inva…