Study on stable torsion length in groups, showing it vanishes in crystallographic groups and providing algorithms for computation.
problem Understanding the stable torsion length in groups, especially in crystallographic and free products of groups.
method Developed linear programming and exact algorithms to compute stable torsion length in free products of groups and finite groups.
result Showed that stable torsion length vanishes in crystallographic groups and provided exact computations for nontrivial examples.
The study reveals conditions for generalized torsion in 3-manifold groups.
problem Conditions for the presence of generalized torsion in 3-manifold fundamental groups.
method Analyzes free products of torsion-free groups and decomposes 3-manifolds.
result Infinitely many toroidal 3-manifolds have generalized torsion elements, while their decomposing pieces do not.
Study builds non-bi-orderable groups without generalized torsion.
problem Constructing non-bi-orderable groups without generalized torsion.
method Constructing specific one-relator groups.
result Demonstrates existence of non-bi-orderable groups without generalized torsion.
Paper finds generalized torsions in non-bi-orderable 3-manifold groups.
problem Finding non-bi-orderable 3-manifold groups.
method Analyzing fundamental groups of once punctured torus bundles.
result Identifies generalized torsions in non-bi-orderable groups.
Study shows torsion homology growth vanishes for certain free-by-cyclic groups.
problem Understanding torsion homology growth in free-by-cyclic groups.
method Analyzing polynomially growing monodromy and showing vanishing homology torsion.
result Integral torsion equals ℓ2-torsion for these groups, verifying a conjecture. Analytic torsion defined for non-compact Lie groups and discrete subgroups.
problem Defining and calculating analytic torsion for non-compact Lie groups and their discrete subgroups.
method Localised analytic torsion and relative analytic torsion defined for Lie groups of type I, using representations and discrete subgroups.
result Relative analytic torsion of (G,Γ) coincides with Lott L2 analytic torsion of a covering space. New findings on generating mapping class groups with specific torsion elements.
problem Understanding the structure of mapping class groups through torsion elements.
method Analyzing the orders of torsion elements required to generate mapping class groups of surfaces of varying genus.
result For specific genera, mapping class groups can be generated by two torsion elements of particular orders.
Homotopy braid groups are shown to be torsion-free.
problem The existence of non-abelian torsion-free quotients of the braid group.
method Diagrammatic theory of welded braids and Artin representation.
result Homotopy braid groups are torsion-free.
Generalizes hyperbolicity to groups with torsion.
problem Hyperbolicity of groups with torsion.
method Introduces bicollapsible 2-complexes to generalize hyperbolicity.
result Many groups act properly and cocompactly on CAT(0) cube complexes.
Topological complexity of hyperbolic groups equals twice their torsion number.
problem Calculating the topological complexity of hyperbolic groups.
method Analyzing finitely generated torsion free hyperbolic groups with a specific formula.
result The topological complexity of a hyperbolic group is twice its torsion number.
One-relator groups with torsion are shown to be coherent.
problem One-relator groups with torsion.
method Demonstrated coherence by showing every finitely generated subgroup is finitely presented.
result One-relator groups with torsion are coherent.
The paper computes torsion invariants for groups acting on complexes.
problem Computing torsion invariants for groups acting on complexes.
method Analyzes residually finite groups acting cocompactly on contractible complexes with specific stabilizers.
result Torsion limits to the torsion of the boundary subcomplex, independent of the chain of subgroups.
The paper classifies 3-manifold groups with specific torsion elements.
problem Classifying 3-manifold groups with generalized torsion elements of order two.
method Analyzing the fundamental groups of 3-manifolds and their conjugates.
result 3-manifold groups with generalized torsion elements of order two have been classified.
New results on homology torsion growth for various groups.
problem Understanding the growth of higher torsion homologies for arithmetic lattices and other groups.
method Quantitative homotopical method called effective rebuilding, constructing small classifying spaces of finite index subgroups.
result Strong asymptotic bounds for the torsion growth in principal congruence subgroups.
The study finds generalized torsion elements in 3-manifolds from specific knot surgeries.
problem Identifying generalized torsion elements in 3-manifolds from knot surgeries.
method Using the JSJ-decomposition of the 3-manifold and bi-orderability properties.
result Existence and properties of generalized torsion elements in specific 3-manifolds.
It is well known that any knot group is torsion-free, but it may admit a generalized torsion element. We show that the knot group of any negative twist knot admits a generalized torsion element. This is a generalization of the same claim for the knot 52, which is the (−2)-twist knot, by Naylor and Rolfsen.
Generic groups can't move spaces but have rich actions.
problem Generic torsion-free groups and their actions.
method Model theoretic forcing
result Generic countable torsion-free groups do not admit nontrivial locally moving actions.
New examples of hyperbolic links with generalized torsion elements found.
problem Finding generalized torsion elements in the fundamental groups of hyperbolic links.
method Analyzing the Weeks manifold, figure-eight sister manifold, and Whitehead sister link to identify generalized torsion elements.
result First examples of hyperbolic links with link groups admitting generalized torsion elements.
Analytic torsion and Reidemeister torsion help show exponential growth of torsion in cohomology.
problem Exponential growth of torsion in the cohomology of arithmetic groups.
method Analytic torsion and Reidemeister torsion, applied to fibered cusp ends of manifolds.
result Exponential growth of torsion in the cohomology of arithmetic groups.
New generalized torsion found in 3-manifolds.
problem Understanding torsion in 3-manifolds.
method Constructing a generalized torsion element.
result Found a new generalized torsion element in a 3-manifold.
Torsion-free connections on G-structures are proven for certain groups.
problem Proving torsion-free connections on G-structures for specific groups. method Classification of admissible groups and construction of connections.
result Torsion-free connections can be locally Levi-Civita connections in a given conformal class.
Groups with specific properties have vanishing ℓ2-Betti numbers.
problem Understanding ℓ2-Betti numbers for certain groups. method Introduced cheap 1-rebuilding property and used structure theorem of Tucker-Drob.
result First ℓ2-Betti numbers vanish for specified groups. We obtain a number of results regarding freeness, quasiconvexity and separability for subgroups of Coxeter groups, Artin groups and one-relator groups with torsion.
New group-theoretic Johnson classes applied to curves with torsion Ceresa classes.
problem Analyzing torsion in Ceresa classes of curves.
method Group-theoretic analogues of Johnson/Morita cocycles applied to pro-l etale fundamental groups of curves.
result Example of a non-hyperelliptic curve with torsion Ceresa class.
The study finds infinitely many hyperbolic 3-manifolds with large rank and generalized torsion elements.
problem Finding hyperbolic 3-manifold groups with large rank and generalized torsion elements.
method Constructing specific hyperbolic 3-manifolds with given properties.
result Infinitely many hyperbolic 3-manifolds with generalized torsion elements of arbitrarily large order.
Study primes dividing torsion in homology of commuting elements in Lie groups.
problem Which primes divide the homology torsion of commuting elements in Lie groups?
method Homotopy decomposition of Hom(Z^m, G)_1, combinatorics of Weyl groups.
result Torsion in homology of Hom(Z^m, G)_1 depends on the Weyl group of G.
Study Whitehead torsions in inertial h-cobordisms.
problem Understanding Whitehead torsions in specific geometric contexts.
method Analyzing inertial h-cobordisms to identify subsets of the Whitehead group.
result Identified various types of Whitehead torsions representing nested subsets of the Whitehead group.
Study on minimal torsion topological generators for mapping class groups of infinite-type surfaces.
problem Minimal topological generating sets of mapping class groups consisting of torsion elements.
method Investigation of minimal topological generating sets for Map(S(n)) consisting entirely of torsion elements, with special attention to involutions. result Minimal topological generating sets for Map(S(n)) consisting of torsion elements are found for various n. We compute K-homology for hyperbolic reflection groups, showing torsion-freeness.
problem Computing K-homology for hyperbolic reflection groups. method Using Baum-Connes conjecture and new algebraic criterion for torsion-freeness.
result Torsion-freeness of K-theory groups for hyperbolic reflection groups. Researchers compute torsion for homology cylinders, proving it a finite-type invariant.
problem Computing finite-type invariants for homology cylinders.
method Non-commutative Reidemeister-Turaev torsion, using K1-group and group ring. result The torsion's reduction is a finite-type invariant of degree d. Constructs equivariant analytic torsion for proper actions on manifolds.
problem Analytic torsion for proper actions on Riemannian manifolds.
method Equivariant construction of Ray-Singer analytic torsion for proper, isometric actions.
result Generalizes earlier results to noncompact conjugacy classes under suitable conditions.
Study characterizes G2-structures on specific Lie groups and identifies harmonic conditions.
problem Characterizing and identifying harmonic G2-structures on almost Abelian Lie groups. method Analyzing left-invariant G2-structures, characterizing torsion forms, and using algebraic conditions. result Established algebraic conditions for harmonic G2-structures and identified admissible torsion classes. We show a relationship between the non-acyclic Reidemeister torsion and a zero of the acyclic Reidemeister torsion for a lambda-regular SU(2) or SL(2, C)-representation of a knot group. Then we give a method to calculate the non-acyclic Reidemeister torsion of a knot exterior. We calculate a new example and investigate…
New findings on Chern flat metrics and their criticality.
problem Understanding critical Hermitian metrics on Chern flat manifolds.
method Analyzing Chern flat manifolds as compact quotients of complex Lie groups and studying their criticality.
result Chern flat metrics on semi-simple Lie groups are torsion-critical and vice versa.
New infinite class of hyperbolic knots with high genus and generalized torsion found.
problem Finding knots with high genus and generalized torsion.
method Demonstrated through (2, 2q+1)-torus knots and their twist families.
result Infinitely many hyperbolic knots with arbitrarily high genus and generalized torsion.
We classify up to coarse equivalence all countable abelian groups of finite torsion free rank. The Q-cohomological dimension and the torsion free rank are the two invariants that give us such classification. We also prove that any countable abelian group of finite torsion free rank is coarsely equivalent to Z^n + H whe…
Researchers identify knot groups with generalized torsion of order two.
problem Understanding knot groups with specific algebraic properties.
method Analyzing knot groups through generalized torsion, unique root property, and Baumslag-Solitar relations.
result Knot groups with generalized torsion of order two are R-groups and $ar{R}$-groups. Study shows how torsion in fundamental groups is related to fiber bundles.
problem Understanding torsion in fundamental groups of fiber bundles.
method Presented a condition for towers of fiber bundles to imply nilpotent subgroups with bounded torsion in their center.
result The index of the subgroup can be bounded in terms of the fibers of the tower.
Given a finite set of r points in a closed surface of genus g, we consider the torsion elements in the mapping class group of the surface leaving the finite set invariant. We show that the torsion elements generate the mapping class group if and only if (g,r)=(2,5k+4) for some integer k.
The paper classifies reversible elements in Seifert-fibered spaces and braid groups.
problem Classifying reversible elements in Seifert-fibered spaces and braid groups.
method Classification of reversible elements in Fuchsian groups, application to Seifert-fibered groups, and analysis of 3-torsion elements.
result Classification and analysis of reversible and 3-torsion elements in Seifert-fibered spaces and braid groups.
We show that a PSL(2;R)-representation of a Fuchsian group induces the asymptotics of the Reidemeister torsion for the Seifert manifold corresponding to the euler class of the PSL(2;R)-representation. We also show that the limit of leading coefficient of the Reidemeister torsion is determined by the euler class of a PS…
The paper proves positivity of a L2-torsion function for certain 3-manifolds.
problem Positivity of a L2-torsion function for 3-manifolds. method Analyzes the representation variety and uses properties of the fundamental group.
result The L2-torsion function is strictly positive for 3-manifolds with infinite fundamental group. Given an L2-acyclic connected finite CW-complex, we define its universal L2-torsion in terms of the chain complex of its universal covering. It takes values in the weak Whitehead group Whw(G). We study its main properties such as homotopy invariance, sum formula, product formula and Poincaré d…
New knot groups found to be bi-orderable using pretzel knots.
problem Understanding bi-orderability of knot groups.
method Applied Mayland's technique to pretzel knots to show their commutator subgroups are residually-torsion-free nilpotent.
result Found new examples of bi-orderable knot groups for non-fibered or alternating knots.
Paper proves homotopy braid group properties over integers and three strands.
problem Understanding homotopy braid group properties.
method Proved linearity over integers and torsion freeness for three strands.
result Homotopy braid group on three strands is torsion free.
Analytic torsion expansions for symmetric and complex homogeneous spaces.
problem Calculating the full asymptotic expansion of analytic torsion for various spaces.
method Explicit calculation and comparison with existing results.
result Explicit full asymptotic expansions for symmetric and complex homogeneous spaces.
Formula connects analytic and Reidemeister torsions for hyperbolic manifolds.
problem Relating analytic and Reidemeister torsions for hyperbolic manifolds.
method Derives a formula relating analytic torsion to Reidemeister torsion of compactified manifolds.
result Formula connects analytic and Reidemeister torsions for hyperbolic manifolds.
Paper solves equations of any length in a specific group.
problem Finding solutions to equations over torsion-free groups of arbitrary length.
method Examined three equations of arbitrary length and showed they have a solution under certain conditions.
result Equations have solutions if two relations among coefficients hold.