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.
Let G be a group hyperbolic relative to a finite collection of subgroups P. Let F be the family of subgroups consisting of all the conjugates of subgroups in P, all their subgroups, and all finite subgroups. Then there is a cocompact model for EFG. This result was known…
We generalize Bonahon's characterization of geometrically infinite torsion-free discrete subgroups of PSL(2, C) to geometrically infinite discrete isometry subgroups in the case of rank 1 symmetric spaces, and, under the assumption of bounded torsion, to the case of negatively pinched Hadamard manifolds. Eve…
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…
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.
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 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.
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.
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. We present a condition for towers of fiber bundles which implies that the fundamental group of the total space has a nilpotent subgroup of finite index whose torsion is contained in its center. Moreover, the index of the subgroup can be bounded in terms of the fibers of the tower. Our result is motivated by the conject…
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.
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…
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.
We show that any one-relator group G=F/⟨⟨w⟩⟩ with torsion is coherent -- i.e., that every finitely generated subgroup of G is finitely presented -- answering a 1974 question of Baumslag in this case.
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. 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…
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.
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 $…
Birman-Lubotzky-McCarthy proved that any abelian subgroup of the mapping class groups for orientable surfaces is finitely generated. We apply Birman-Lubotzky-McCarthy's arguments to the mapping class groups for non-orientable surfaces. We especially find a finitely generated group isomorphic to a given torsion-free sub…
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.
In this paper, we generalize Bonahon's characterization of geometrically infinite torsion-free discrete subgroups of PSL(2, C) to geometrically infinite discrete subgroups Γ of isometries of negatively pinched Hadamard manifolds X. We then generalize a theorem of Bishop to prove that every discrete geome…
We associate cube complexes called completions to each subgroup of a right-angled Coxeter group (RACG). A completion characterizes many properties of the subgroup such as whether it is quasiconvex, normal, finite-index or torsion-free. We use completions to show that reflection subgroups are quasiconvex, as are one-end…
For d=2n+1 a positive odd integer, we consider sequences of arithmetic subgroups of SO_0(d,1) and Spin(d,1) yielding corresponding hyperbolic manifolds of finite volume and show that, under appropriate and natural assumptions, the torsion of the associated cohomology groups grows exponentially.
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 …
If X is an orientable, strongly minimal PD4-complex and π1(X) has one end then it has no nontrivial locally-finite normal subgroup. Hence if π is a 2-knot group then (a) if π is virtually solvable then either π has two ends or π≅Φ, with presentation ⟨a,t∣ta=a2t⟩, or π is torsion-f…
Study of Hermitian structures on Lie groups with 2D commutator subgroups.
problem Classifying Hermitian structures on Lie groups with specific commutator subgroups.
method Explicit classification of Type I and Type II structures, computation of Bismut connections, and examples of Kahler structures.
result Classification of Kahler structures within Type I and Type II structures.
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. We prove that both the hyperelliptic mapping class group and the extended hyperelliptic mapping class group are generated by two torsion elements. We also compute the index of the subgroup of the hyperelliptic mapping class group which is generated by involutions and we prove that the extended hyperelliptic mapping cla…
The canonical-type connection on the almost contact manifolds with B-metric is constructed. It is proved that its torsion is invariant with respect to a subgroup of the general conformal transformations of the almost contact B-metric structure. The basic classes of the considered manifolds are characterized in terms of…
Locally connected boundaries proven for relatively hyperbolic groups.
problem Proving local connectedness of boundaries for relatively hyperbolic groups.
method Using a group pair (Γ,P) that is relatively one ended, and removing restrictions on cardinality and peripheral subgroups. result The Bowditch boundary of (Γ,P) is locally connected.