The paper explores continuous inverse ambiguous functions on various Lie groups.
problem Existence of continuous inverse ambiguous functions on Lie groups.
method Investigation of continuous inverse ambiguous functions on specific Lie groups.
result Existence of continuous inverse ambiguous functions on various Lie groups.
This paper provides a structural decomposition for extended function groups.
problem No existing literature states a structural decomposition for extended function groups.
method Uses the Klein-Maskit combination theorems.
result States and proves a structural decomposition for extended function groups.
Study on homological Dehn functions of groups of type FP2.
problem Understanding the homological Dehn functions of groups of type FP2. method Proved foundational results, studied homological Dehn functions of Leary's groups, and provided methods to obtain groups with specific homological Dehn functions.
result Found groups of type FP2 with quartic homological Dehn function and unsolvable word problem. The paper shows how coarse embeddings affect homological Dehn functions.
problem Characterizing groups with coarse embeddings into hyperbolic groups.
method Demonstrates a coarse embedding theorem for homological filling functions.
result Characterizes groups with coarse embeddings into hyperbolic groups of geometric dimension 2.
The study explores the Dehn functions of Kähler groups and their properties.
problem Which functions can arise as Dehn functions of Kähler groups?
method Analyzes examples of Kähler groups with various Dehn functions and proves the existence of a Kähler group with a cubic bounded Dehn function.
result There exists a Kähler group with a cubic bounded Dehn function and an exponential upper bound.
Groups satisfy linear surface isoperimetric functions.
problem Isoperimetric functions for surface diagrams in hyperbolic groups.
method Analyzing word-hyperbolic groups and their surface diagrams.
result Linear isoperimetric functions for all surface types in hyperbolic groups.
Study Kleinian groups using orbital functions and heat kernels.
problem Understanding Kleinian groups through orbital functions and heat kernels.
method Using the heat kernel approach developed in \cite{artmoiheatcounting1}.
result Developed a method to study Kleinian groups using orbital functions and heat kernels.
Precise computations of Dehn functions for subgroups of free group products.
problem Computing precise Dehn functions for subgroups of direct products of free groups.
method Analyzing specific subgroups and using algebraic methods to compute Dehn functions.
result Quartic and quadratic Dehn functions for specific subgroups of free group products.
Handlebody groups have exponential Dehn functions except for genus 2.
problem Understanding the complexity of handlebody groups through their Dehn functions.
method Analyzing the Dehn functions of handlebody groups of different genera.
result Handlebody groups of genus 3 and above have exponential Dehn functions, while genus 2 has a quadratic one.
Study length functions on various groups and prove homomorphisms to finite groups.
problem Understanding length functions on different types of groups.
method Analyzing length functions on Lie groups, Gromov hyperbolic groups, arithmetic subgroups, matrix groups, and Cremona groups.
result Prove that homomorphisms to certain groups must have finite images.
New CAT(0) groups show superexponential subgroup Dehn functions.
problem Understanding subgroups of CAT(0) groups with superexponential Dehn functions.
method Construction of specific 4- and 6-dimensional CAT(0) groups.
result Found subgroups with Dehn functions exp(n)(xm) and exp(n)(xα). New method groups similar functional covariates for better modeling.
problem Analyzing functional covariates with similar shapes.
method Coefficient shape alignment regularization approach.
result True grouping structure can be accurately identified under certain conditions.
New biharmonic functions created on Lie groups.
problem Constructing explicit biharmonic functions on Lie groups.
method Developed a new scheme for constructing complex-valued biharmonic functions on Riemannian Lie groups.
result Manufactured infinite series of new solutions on SU(n) and showed applicability to SO(n) and Sp(n). Researchers determine Dehn functions of specific nilpotent groups.
problem Understanding the Dehn functions of central products of nilpotent groups.
method Analyzing families of filiform and Lie groups to determine Dehn functions.
result Confirms conjecture and provides evidence for lower Dehn functions in central products.
In the paper `Automorphic functions for a Whitehead-complement group', [Osaka J Math 43 (2006) 63-77] Matsumoto, Nishi and Yoshida constructed automorphic functions on real 3-dimensional hyperbolic space for a Kleinian group called the Whitehead-link-complement group. For a Kleinian group (of the first kind), no automo…
New method constructs explicit p-harmonic functions on Lie groups.
problem Constructing proper p-harmonic functions on Lie groups. method Employing complex isoparametric functions to devise a general method.
result First explicit proper p-harmonic functions on Rm⋉Rn and Rm⋉H2n+1. New filling functions for groups with coefficients show different asymptotic behavior.
problem Difficulty in filling loops with surfaces in Cayley graphs.
method Defining homological filling functions with coefficients and proving their differences.
result Filling functions for n-cycles with coefficients in different groups have distinct asymptotic behavior. We prove super-quadratic lower bounds for the growth of the filling area function of a certain class of Carnot groups. This class contains groups for which it is known that their Dehn function grows no faster than n2logn. We therefore obtain the existence of (finitely generated) nilpotent groups whose Dehn functio…
We classify Dehn functions of Bestvina-Brady groups.
problem Understanding the complexity of Bestvina-Brady groups.
method Explicit criteria on defining graphs to determine Dehn function degree.
result Explicitly classify the Dehn functions of Bestvina-Brady groups.
Derives representations invariant under crystallographic groups for functions.
problem Representing and learning functions invariant under crystallographic groups.
method Derives linear and nonlinear representations of functions invariant under crystallographic groups.
result Derives orthonormal crystallographically invariant basis functions and embedding maps.
We bound the higher-order Dehn functions and other filling invariants of certain Carnot groups using approximation techniques. These groups include the higher-dimensional Heisenberg groups, jet groups, and central products of two-step nilpotent groups. Some consequences of this work are a construction of groups with ar…
In this paper we prove trace formulae for the Reidemeister number of a group endomorphism. This result implies the rationality of the Reidemeister zeta function in the following cases: the group is a direct product of a finite group and a finitely generated Abelian group; the group is finitely generated, nilpotent and …
Homological stability fails for Cremona groups, rational varieties, and function fields.
problem Homological stability in Cremona groups fails in both possible ways.
method Explained the failure of homological stability for Cremona groups.
result Homological stability fails for Cremona groups in both possible ways.
Constructs explicit p-harmonic functions on specific Lie groups.
problem Finding explicit p-harmonic functions on a specific class of Lie groups.
method Constructs explicit p-harmonic functions on rank-one Lie groups of Iwasawa type.
result Proves existence of proper p-harmonic functions on these groups.
The article studies groups generated by products and wreath products, focusing on first Betti numbers of Morse function orbits.
problem Calculating first Betti numbers of orbit groups generated by products and wreath products.
method Analyzes algebraic properties of specific groups G, proving ranks of center and quotient by commutator subgroup coincide. result The rank of the quotient by commutator subgroup is a first Betti number of the orbit of Morse function.
In this paper, we consider the formal power series whose n-th coefficient is the number of copies of a given finite graph in the ball of radius n centred at the identity element in the Cayley graph of a finitely generated group and call it the growth function. Epstein, Iano-Fletcher and Uri Zwick proved that the growth…
Morse inequalities for noncompact manifolds with group action.
problem Establishing inequalities for noncompact manifolds with group action.
method Using L2-Betti numbers and functions describing critical points. result Morse inequalities given in terms of L2-Betti numbers and group functions. New growth rate identified for quaternionic Heisenberg group filling functions.
problem Identifying the growth rate of quaternionic Heisenberg group filling functions.
method Analyzing the growth of filling volume functions in the quaternionic Heisenberg group up to dimension n+1.
result Strictly faster growth rate identified for dimension n+1 compared to Euclidean space.
Lectures on complex hyperbolic spaces and their groups.
problem Understanding interactions between complex hyperbolic spaces and discrete groups.
method Discussion of function theory and discrete group theory.
result Interactions between complex hyperbolic spaces and discrete groups.
The paper generalizes polynomial functions on Lie groups and their properties.
problem Generalizing polynomial functions on Lie groups and understanding their properties.
method Generalizing the notion of polynomial functions and horizontally affine maps on Lie groups.
result S-polynomial functions are equivalent to polynomial functions in the sense of Leibman in connected nilpotent Lie groups.
Paper refines generating function for 2-bridge knot groups.
problem Determining the number of epimorphisms between 2-bridge knot groups.
method Refined generating function considering genus and crossing number.
result Improved formula for epimorphisms between 2-bridge knot groups.
New groups with distinct Dehn functions and properties.
problem Finding groups with different Dehn functions.
method Created specific Lie groups and Carnot graded groups.
result Groups with uniform lattices have different asymptotic cones and Dehn functions.
Characterizes functions in Carnot groups of step 2.
problem Understanding intrinsic Lipschitz functions in Carnot groups.
method Characterization via intrinsic distributional gradients.
result Characterization of locally intrinsic Lipschitz functions in Carnot groups of step 2.
Explains connections between free groups and positive definite functions.
problem Understanding positive definite functions on free groups.
method Expository survey of known results and new perspectives.
result New relationships between free groups and positive definite functions.
Study of Morse functions with constraints and their bordism groups.
problem Interpolating between Morse and generic functions' bordism groups.
method Elimination of cusps, Stein factorization, two-index theorem, handle extension theorem.
result Constrained bordism groups are related to connective bordism.
Group-invariant neural networks improve approximation accuracy for symmetric functions.
problem Improving approximation accuracy for symmetric functions using neural networks.
method Investigates the generalization error of group-invariant neural networks within the Barron framework.
result Group invariance introduces a factor δ that can significantly improve approximation accuracy when it is small.
Researchers describe Casimir functions for 3- and 4-step nilpotent Lie groups.
problem Understanding Casimir functions for free nilpotent Lie groups of steps 3 and 4.
method Construction of Casimir functions for free nilpotent Lie groups of steps 3 and 4.
result For 3-step groups, coadjoint orbits are fully described as affine subspaces or direct products of quadrics.
Filling invariants are measurements of a metric space describing the behaviour of isoperimetric inequalities. In this article we examine filling functions and higher divergence functions. We prove for a class of stratified nilpotent Lie groups that in the low dimensions the filling functions grow as fast as the ones of…
Paper describes invariants of slice regular functions' automorphism group.
problem Understanding invariants of slice regular functions' automorphism group.
method Analyzes automorphism group of slice regular functions over Clifford algebras.
result Describes invariants of the automorphism group of slice regular functions.
Counting orbits for Anosov groups with specific functionals.
problem Counting orbits for relatively Anosov groups with linear functionals.
method Equidistribution results and previous counting results for periods.
result Generalization of earlier work on Anosov groups.
The HOMFLYPT polynomial yields a positive function for a specific group.
problem Constructing representations of the oriented Thompson group.
method Using the HOMFLYPT polynomial as a link invariant.
result The HOMFLYPT polynomial defines a positive definite function of the oriented Thompson group.
Formula for signature of handlebody bundles, interpreting cohomology.
problem Signature of handlebody bundles and their monodromy.
method Explicit formula using homological monodromy.
result Cobounding function of Meyer's signature cocycle on handlebody group.
We introduce a new invariant of bipartite chord diagrams and use it to construct the first examples of groups with Dehn function n2logn and other small Dehn functions. Some of these groups have undecidable conjugacy problem.
We construct a group (an HNN extension of a free group) with polynomial isoperimetric function, linear isodiametric function and non-simply connected asymptotic cones.
The paper classifies pairs of Morse functions under different groups.
problem Classifying pairs of Morse functions in general position.
method Analysis of pairs of Morse functions under various groups.
result Classification of generic pairs of Morse functions, including quotients.
New p-harmonic and harmonic morphisms found on Lie groups.
problem Constructing explicit p-harmonic and harmonic morphisms on Lie groups.
method Using the method of eigenfamilies to construct explicit complex-valued p-harmonic functions and harmonic morphisms.
result Explicit complex-valued p-harmonic functions and harmonic morphisms constructed on non-compact classical Lie groups.
A homogeneous nilpotent Lie group has a scaling automorphism determined by a grading of its Lie algebra. Many proofs of upper bounds for the Dehn function of such a group depend on being able to fill curves with discs compatible with this grading; the area of such discs changes predictably under the scaling automorphis…
Classifies geodetically convex sets and functions on Heisenberg group.
problem Characterizing geodetically convex sets and functions in the Heisenberg group.
method Classification through mathematical analysis.
result Geodetically convex sets and functions defined on Heisenberg group Hn classified.