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. Study shows conjugacy of torsion in genus 2 surfaces.
problem Torsion elements in mapping class groups of surfaces.
method Proved congruence subgroup property for centralizers of finite subgroups.
result Torsion elements in surfaces of genus ≤ 2 are conjugacy distinguished.
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.
We obtain a number of results regarding freeness, quasiconvexity and separability for subgroups of Coxeter groups, Artin groups and one-relator groups with torsion.
Paper proves non-triviality of Johnson kernel torsion subgroup.
problem Non-triviality of the torsion subgroup of the abelianized Johnson kernel.
method Action of mapping class group on Malcev Lie algebra, diagrammatic techniques.
result Proves non-triviality of the torsion subgroup with a purely 2-dimensional proof.
Study homology growth in nonpositive curvature spaces, finding examples of torsion.
problem Understanding homology growth in nonpositive curvature spaces.
method Computing mod p homology growth of right-angled Artin groups and closed locally CAT(0) manifolds.
result Homology torsion grows exponentially in the index of subgroups, contradicting rational homology growth.
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.
New findings on infinite subgroups in negatively curved spaces.
problem Characterizing infinite discrete isometry subgroups in negatively pinched Hadamard manifolds.
method Generalization of Bonahon's characterization to negatively pinched Hadamard manifolds.
result Every geometrically infinite isometry subgroup has a continuum of nonconical limit points.
New method for compact space of group subgroups, useful in hyperbolic groups.
problem Compactifying the classifying space for parabolic subgroups in relatively hyperbolic groups.
method Exploiting the notion of dismantlability to handle torsion.
result A cocompact model for the family of subgroups in relatively hyperbolic groups.
In this paper we prove that for suitable sequences of congruence subgroups of Bianchi groups, including the standard exhaustive sequences of a congruence subgroup, and even symmetric powers of the standard representation of Sl_2(C) the size of the torsion part in the first homology grows exponentially. This extends res…
Extends Nielsen realization to infinite-type surfaces, classifying torsion elements and topological groups.
problem Realizing finite subgroups of mapping class groups on infinite-type surfaces.
method Extending Kerckhoff's result to infinite-type surfaces, using hyperbolic metrics and topological group properties.
result Compact subgroups of mapping class groups are finite, and locally compact subgroups are discrete.
We prove that every finitely generated Kleinian group that contains a finite, non-cyclic subgroup either is finite or virtually free or contains a surface subgroup. Hence, every arithmetic Kleinian group contains a surface subgroup.
We study knots of order 2 in the grope filtration $\{\G_h\}$ and the solvable filtration $\{\F_h\}$ of the knot concordance group. We show that, for any integer n≥4, there are knots generating a Z2∞ subgroup of $\G_n/\G_{n.5}$. Considering the solvable filtration, our knots generate a Z2∞ subgro…
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 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.
Study shows torsion normal generators in non-orientable surface mapping class groups.
problem Understanding normal generators of mapping class groups for non-orientable surfaces.
method Analyzes periodic elements and their normal closures in non-orientable surfaces.
result Provides conditions for normal generation and examples of elements.
Magnitude homology reveals that graphs can have torsion subgroups.
problem Understanding torsion in magnitude homology of graphs.
method Analysis of magnitude homology defined by Hepworth and Willerton.
result Torsion of any prime order can appear in graphs' magnitude homology.
The paper classifies fixed subgroups of endomorphisms in free-abelian times surface groups.
problem Characterizing fixed subgroups of endomorphisms in specific group structures.
method Study of endomorphisms, classification of fixed subgroups, and equivalent conditions for end-fixed subgroups.
result Complete classification of fixed subgroups in free-abelian times surface groups.
The study bounds the growth of torsion in homology and counts non-commensurable hyperbolic manifolds.
problem Bounding the growth of torsion in homology of hyperbolic manifolds.
method Analyzes the growth of torsion subgroups in homology and counts non-commensurable manifolds.
result The number of non-commensurable closed hyperbolic manifolds grows exponentially with diameter.
Starting from the results in math.DG:1212.3161 we prove that for a given Bianchi group, certain natural coefficent modules and a lot of sequences of congruence subgroups of the size of the torsion subgroup of the first homology grows exponentially with the index (we give an explicit rate). We also prove limit multiplic…
Brin-Thompson groups have new properties for n>=2.
problem Characterize subgroups of Brin-Thompson groups.
method Analyzing torsion and subgroup structure.
result Brin-Thompson groups nV for n≥2 contain continuum many copies of Q. 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.
Proves existence of certain subgroups in hyperbolic groups.
problem Existence of weakly malnormal quasiconvex subgroups in hyperbolic groups.
method Proof of existence in nonelementary hyperbolic groups.
result Existence of weakly malnormal, virtually free, quasiconvex subgroups in hyperbolic groups.
We obtain a number of finiteness results for groups acting on Gromov-hyperbolic spaces. In particular we show that a torsion-free locally quasiconvex hyperbolic group has only finitely many conjugacy classes of n-generated one-ended subgroups. We also show that the rank problem is solvable for the class of torsion-fr…
Non-normal subgroups of certain groups grow homologically exponentially.
problem Homological torsion growth in non-normal subgroups of specific groups.
method Proving exponential growth of homological torsion in a sequence of non-normal subgroups.
result Exponential homological torsion growth in a sequence of non-normal subgroups.
This paper restricts torsion subgroups in homology groups of Reeb spaces.
problem Understanding the structure of homology groups of Reeb spaces of fold maps.
method Explicitly studying the algebraic constraints on homology groups of Reeb spaces constructed by surgery operations.
result Explicit strong restrictions on the torsion subgroups of homology groups.
In this paper we prove that for a fixed neat principal congruence subgroup of a Bianchi group the order of the torsion part of its second cohomology group with coefficients in an integral lattice associated to the m-th symmetric power of the standard representation of SL_2(C) grows exponentially in m^2. We give upper a…
Torsion in cohomology grows exponentially for certain arithmetic hyperbolic manifolds.
problem Growth of torsion in cohomology of arithmetic hyperbolic manifolds.
method Study of sequences of arithmetic subgroups of SO_0(d,1) and Spin(d,1) yielding hyperbolic manifolds.
result Torsion in cohomology grows exponentially under natural assumptions.
Torelli group cannot be realized as area-preserving homeomorphisms.
problem Realization of the Torelli group as area-preserving homeomorphisms.
method Analysis of the Torelli group and its relationship with homeomorphisms.
result The Torelli group has no realization inside the area-preserving homeomorphisms.
Study of a torsion class in mapping class group's cohomology.
problem Understanding torsion in mapping class groups' cohomology.
method Analyzing the inclusion of mapping class groups into circle homeomorphisms and studying the pullback of Euler classes.
result Proves the existence of a torsion class in cohomology, generating a cyclic subgroup of order multiple of 4g(2g+1)(2g−1). 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.
Abstract: Study of 2-knot groups with restrictions on normal subgroups.
problem Characterizing 2-knot groups based on their normal subgroups.
method Analyzing PD_4-complexes and using properties of π_1(X).
result Characterization of 2-knot groups based on their normal subgroups.
We investigate the rank gradient and growth of torsion in homology in residually finite groups. As a tool, we introduce a new complexity notion for generating sets, using measured groupoids and combinatorial cost. As an application we prove the vanishing of the above invariants for Farber sequences of subgroups of righ…
Plante-Thurston proved that every nilpotent subgroup of $\Diff^2(S^1)$ is abelian. One of our main results is a sharp converse: $\Diff^1(S^1)$ contains every finitely-generated, torsion-free nilpotent group.
New generators found for twist subgroup of nonorientable surfaces.
problem Identifying generators for twist subgroup of nonorientable surfaces.
method Analyzing mapping class group of nonorientable surfaces of genus g≥13. result Twist subgroup can be generated by two involutions and an element of order g or g−1. The study examines conditions for symmetric and alternating subgroups in mapping class groups of surfaces.
problem Conditions for torsion elements to generate symmetric or alternating subgroups.
method Analyzes mapping class groups of surfaces, derives necessary and sufficient conditions for conjugates of torsion elements to generate symmetric or alternating subgroups.
result Symmetric or alternating subgroups cannot contain irreducible mapping classes and hyperelliptic involutions.
Study non-orientable surfaces, find finitely generated abelian subgroups.
problem Characterize abelian subgroups in non-orientable surfaces.
method Apply Birman-Lubotzky-McCarthy's arguments to non-orientable surfaces.
result Find a finitely generated group isomorphic to a given subgroup.
Let Hn be the hyperbolic n-space with n≥2. Suppose that Γ<IsomHn is a discrete, torsion free subgroup and a is a point in the domain of discontinuity Ω(Γ). Let p be the projection map from Hn to the quotient manifold M=Hn/Γ. In this paper we prove that there exists an open neighborhood U of $…
This paper proves bi-orderability of two-bridge link groups.
problem Bi-orderability of two-bridge link groups.
method Modified graph theoretic construction of Hirasawa and Murasugi to understand Alexander subgroups.
result Bi-orderability of a large family of two-bridge link groups.
This paper solves Yang-Baxter cohomology for cyclic biquandles.
problem Yang-Baxter cohomology of cyclic biquandles.
method Completely determined free parts and computed torsion subgroups of homology groups.
result Upper bounds for torsion orders in higher dimensional homology groups.
Improved homological dimension for certain subgroups in Lie groups.
problem Determining homological dimensions of discrete subgroups in Lie groups.
method Using recent results and properties of injectivity radius, the homological dimension gap is calculated.
result Infinite volume torsion-free subgroups of higher rank Lie groups have a homological dimension gap of at least 1/8 of the real rank.
The paper extends geometric finiteness to discrete subgroups of negatively pinched Hadamard manifolds.
problem Characterizing geometrically infinite discrete subgroups of negatively pinched Hadamard manifolds.
method Generalizing Bonahon's characterization and proving a theorem of Bishop's extension.
result Every discrete geometrically infinite isometry subgroup has a set of nonconical limit points of cardinality continuum.
New techniques reveal subgroup properties in Coxeter groups.
problem Characterizing and understanding subgroups of right-angled Coxeter groups.
method Using cube complexes and Stallings-like techniques to study subgroups.
result Reflection and one-ended subgroups are quasiconvex.
Extends growth properties of hyperbolic groups to their extensions.
problem Quantifying subgroup alternatives in group laws.
method Develops a framework for preserving exponential growth in extensions of hyperbolic groups.
result Automorphism groups of certain hyperbolic and Artin groups have locally uniform exponential growth.
Study 2-complexes' homology properties and torsion growth.
problem Quantitative connections between 1-cycle filling inequalities and homology complexities.
method Geometric lower bounds on first homology of finite covers.
result Geometric lower bound on first homology size of finite covers.
We study the growth of the order of torsion subgroups of the homology in a tower of finite abelian coverings. In particular, we prove that it is exponential for when the tower converges to the maximal free abelian cover of a link complement when the first nonzero Alexander polynomial has positive logarithmic Mahler mea…
We consider the unique Hermitian connection with totally skew-symmetric torsion on a Hermitian manifold. We prove that if the torsion is parallel and the holonomy is Sp(n)U(1), considered as a subgroup of U(2n) x U(1), then the manifold is locally isomorphic to the twistor space of a quaternionic Kaehler manifold with …
Study on torsion in homology of Torelli group for surfaces.
problem Torsion in homology of Torelli group for surfaces.
method Analysis of abelian cycles and Dehn twists.
result Subgroup of homology generated by abelian cycles is a finite-dimensional Z/2Z-vector space for k=2 and g≥4.