arXiv research

Based mainly on examples of interest in mechanics, we define the notion of a polite group action. One may view this as not only trying to give a more general notion than properness of a group action, but also to more fully understand the role of invariant functions in describing just about everything of interest in red…

In this paper we introduce distinct approaches to loop braid groups, a generalisation of braid groups, and unify all the definitions that have appeared so far in literature, with a complete proof of the equivalence of these definitions. These groups have in fact been an object of interest in different domains of mathem…

We investigate the combinatorial and geometric properties of automorphism groups of universal right-angled Coxeter groups, which are the automorphism groups of free products of copies of Z_2. It is currently an open question as to whether or not these automorphism groups have non-positive curvature. Analogous to Outer …

Invites geometers to Garside theory for mapping class groups.

Survey on Coxeter groups for Lie group examples.

The purpose of this paper is describe Lagrangian Mechanics for constrained systems on Lie algebroids, a natural framework which covers a wide range of situations (systems on Lie groups, quotients by the action of a Lie group, standard tangent bundles...). In particular, we are interested in two cases: singular Lagrangi…

Origamis described using Schottky groups for surfaces of genus g ≥ 1.

We give a survey of some known results and of the many open questions in the study of generic phenomena in geometrically interesting groups.

We establish an interesting connection between Morin singularities and stable homotopy groups of spheres. We apply this connection to computations of cobordism groups of certain singular maps. The differentials of the spectral sequence computing these cobordism groups are given by the composition multiplication in the …

This article is about Artin's braid group and its role in knot theory. We set ourselves two goals: (i) to provide enough of the essential background so that our review would be accessible to graduate students, and (ii) to focus on those parts of the subject in which major progress was made, or interesting new proofs of…

We study various covering spectra for complete noncompact length spaces with universal covers (including Riemannian manifolds and the pointed Gromov Hausdorff limits of Riemannian manifolds with lower bounds on their Ricci curvature). We relate the covering spectrum to the (marked) shift spectrum of such a space. We de…

New bandit problem for finding best group of arms with worst mean reward.

From a simple observation about a construction of Thurston, we derive several interesting facts about subgroups of the mapping class group generated by two positive multi-twists. In particular, we identify all configurations of curves for which the corresponding groups fail to be free, and show that a subset of these d…

In this article we survey, and make a few new observations about, the surprising connection between sub-monoids of mapping class groups and interesting geometry and topology in low-dimensions.

This paper is the second part of a two-part study of an elementary functorial construction of tesselated surfaces from finite groups. This elementary construction was discussed in the first part and generally results in a large collection of tesselated surfaces per group, for example when the construction is applied to…

The paper proves vanishing bounded cohomology for various groups.

Motivated by the interesting and yet scattered developments in representation theory of Banach-Lie groups, we discuss several functional analytic issues which should underlie the notion of infinite-dimensional reductive Lie group: norm ideals, triangular integrals, operator factorizations, and amenability.

Survey on invariant conformal Killing forms on Lie groups.

One of the most interesting questions about a group is if its word problem can be solved and how. The word problem in the braid group is of particular interest to topologists, algebraists and geometers, and is the target of intensive current research. We look at the braid group from a topological point of view (rather …

We show that surface groups are flexibly stable in permutations. This is the first non-trivial example of a non-amenable flexibly stable group. Our method is purely geometric and relies on an analysis of branched covers of hyperbolic surfaces. Along the way we establish a quantitative variant of the LERF property for s…

We construct quasi-isometric embeddings from right-angled Artin groups into the outer automorphism group of a free group. These homomorphisms are in analogy with those constructed in \cite{CLM}, where the target group is the mapping class group of a surface. Toward this goal, we develop tools in the free group setting …

In this paper we create many examples of hyperbolic groups with subgroups satisfying interesting finiteness properties. We give the first examples of subgroups of hyperbolic groups which are of type $FP_2$ but not finitely presented. We give uncountably many groups of type $FP_2$ with similar properties to those subgro…

MRIF models dynamic user interests at multiple temporal-ranges.

Study Kähler groups from orbifold compactifications of curve moduli.

We investigate the existence of homotopy comoment maps (comoments) for high-dimensional spheres seen as multisymplectic manifolds. Especially, we solve the existence problem for compact effective group actions on spheres and provide explicit constructions for such comoments in interesting particular cases.

Fast algorithm for braid group Hecke representation, applied to knot invariants.

Based on Lie group method, potential symmetry and invariant solutions for generalized quasilinear hyperbolic equations are studied. To obtain the invariant solutions in explicit form, we focus on the physically interesting situations which admit potential symmetries. Then by using the partial Lagrangian approach, we fi…

Outer automorphisms of free products are represented by CTs.

The abstract investigates how volume ratios relate to curvature in geometric surfaces.

The reflection length of an element of a Coxeter group is the minimal number of conjugates of the standard generators whose product is equal to that element. In this paper we prove the conjecture of McCammond and Petersen that reflection length is unbounded in any non-affine Coxeter group. Among the tools used, the con…

Paper proves non-zero generalization boost for equivariant models.

Following previous work of the second author, we establish more properties of groups of circle homeomorphisms which admit invariant laminations. In this paper, we focus on a certain type of such groups-so-called pseudo-fibered groups, and show that many 3-manifold groups are examples of pseudo-fibered groups. We then p…

The symmetries of two-dimensional supersymmetric sigma models on target spaces with covariantly constant forms associated to special holonomy groups are analysed. It is shown that each pair of such forms gives rise to a new one, called a Nijenhuis form, and that there may be further reductions of the structure group. I…

We study subgroups of fundamental groups of real analytic closed 4-manifolds with nonpositive sectional curvature. In particular, we are interested in the following question: if a subgroup of the fundamental group is not virtually free abelian, does it contain a free group of rank two ? The technique involves the theor…

We describe a case of an interplay between human and computer proving which played a role in the discovery of an interesting mathematical result. The unusual feature of the use of computers here was that a computer generated but human readable proof was read, understood, generalized and abstracted by mathematicians to …

We calculate the singular homology and Čech cohomology groups of the Harmonic archipelago. As a corollary, we prove that this space is not homotopy equivalent to the Griffiths space. This is interesting in view of Eda's proof that the first singular homology groups of these spaces are isomorphic.

The paper explores metrics on tree moduli spaces and a new topological group.

This paper is the first part in a 2 part study of an elementary functorial construction from the category of finite non-abelian groups to a category of singular compact, oriented 2-manifolds. After a desingularization process this construction results in a collection of compact, connected, oriented tesselated smooth su…

New groups act on cube complexes without compact cubulation.

The purpose of this study was to build a customer selection model based on 20 dimensions, including customer codes, total contribution, assets, deposit, profit, profit rate, trading volume, trading amount, turnover rate, order amount, withdraw amount, withdraw rate, process fee, process fee submitted, process fee retai…

In this article we raise some new questions about positive definite functions on free groups, and explain how these are related to more well-known questions. The article is intended as a survey of known results that also offers some new perspectives and interesting observations; therefore the style is expository.

A Lie group $G$ endowed with a left invariant Riemannian metric $g$ is called Riemannian Lie group. Harmonic and biharmonic maps between Riemannian manifolds is an important area of investigation. In this paper, we study different aspects of harmonic and biharmonic homomorphisms between Riemannian Lie groups. We show t…

In this note we study the problem of conformally flat structures bounding conformally flat structures and show that the eta invariants give obstructions. These lead us to the definition of an abelian group, the conformal cobordism group, which classifies the conformally flat structures according to whether they bound (…

Owners of a web-site are often interested in analysis of groups of users of their site. Information on these groups can help optimizing the structure and contents of the site. In this paper we use an approach based on formal concepts for constructing taxonomies of user groups. For decreasing the huge amount of concepts…

Study on embedding tree products into groups, distinguishing them.

New hyperbolic groups exhibit unusual finiteness properties.

The class of acylindrically hyperbolic groups, which are groups that admit a certain type of non-elementary action on a hyperbolic space, contains many interesting groups such as non-exceptional mapping class groups and $\operatorname{Out}(\mathbb F_n)$ for $n\geq 2$. In such a group, a generalized loxodromic element i…

We introduce a class of objects which we call 'affine surfaces'. These provide families of foliations on surfaces whose dynamics we are interested in. We present and analyze a couple of examples, and we define concepts related to these in order to motivate several questions and open problems. In particular we generalis…