Proves incoherence of free-by-free and surface-by-free groups, solving two problems.
problem Proving incoherence of free-by-free and surface-by-free groups.
method Semidirect product construction and virtual algebraic fibering analysis.
result Proves incoherence of free-by-free and surface-by-free groups, answering a question posed by J. Hillman.
The paper creates knot invariants using free groups.
problem Invariants of free knots (virtual knots).
method Constructing invariants valued in free groups.
result Series of invariants for free knots.
The study shows that certain groups can be uniquely identified by their finite abelian summands.
problem Identifying groups based on their finite abelian summands.
method Analyzing hyperbolic groups as graphs of free groups with cyclic edge groups.
result Free products of free and surface groups are profinitely rigid.
The pants graph of a free group is constructed and studied.
problem Understanding the structure of free groups through graph theory.
method Developed a pants graph and studied its properties.
result The pants graph of a free group is connected and unbounded.
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.
Elementarily free groups are the finitely generated groups with the same elementary theory as free groups. We prove that elementarily free groups are subgroup separable, answering a question of Zlil Sela.
Proves almost profinite rigidity for certain free-by-cyclic groups.
problem Profinite rigidity of free-by-cyclic groups.
method Analyzes ranks of fibres, characteristic polynomials, and stretch factors of monodromies.
result Generic free-by-cyclic groups are almost profinitely rigid.
Study shows top homology group isn't dualizing module for automorphism group of free groups.
problem Identifying the dualizing module for automorphism group of free groups.
method Analyzing the top homology group of the free factor complex.
result The top homology group is not the dualizing module for extAut(Fn), especially for n=5. Triangle Artin groups split as graphs of free groups under specific conditions.
problem Conditions for triangle Artin groups to split as graphs of free groups.
method Graph of free groups analysis and poly-free property proof.
result Triangle Artin groups split as graphs of free groups if and only if labels are greater than 5 and even.
Study on profinite rigidity of direct products of free and surface groups.
problem Profinite rigidity of direct products in residually free groups.
method Rank gradients of pro-p groups, virtual homology of limit groups. result Direct products of free and surface groups are profinitely rigid.
The study proves conjecture for specific Artin groups.
problem Proving conjecture about Artin groups' properties.
method Analyzing Artin groups associated to triangle-free graphs and cones over square-free bipartite graphs.
result Proves conjecture for specific Artin groups.
The study characterizes subgroups of mapping tori of free groups.
problem Characterizing subgroups of mapping tori of free groups.
method Using canonical maximal sub-mapping tori and relative hyperbolicity.
result Characterizes locally quasi-convex hyperbolic groups.
New proof and description of commutator subgroups for free and surface groups.
problem Understanding commutator subgroups of free and surface groups.
method Geometric proof and representation-theoretic description.
result New free generating sets and structure descriptions for commutator subgroups.
In [V.O. Manturov, Non-reidemeister knot theory and its applications in dynamical systems, geometry, and topology, arxiv:1501.05208] the first named author gave the definition of k-free braid groups Gnk. Here we establish connections between free braid groups, classical braid groups and free groups: we describe e…
Corrects a 1998 proof about free factors of free groups.
problem Verifying the proof of free factors in free groups.
method Analyzes the geometric realization of free factors.
result Geometric realization is homotopy equivalent to a wedge of spheres.
New groups found that don't virtually algebraically fiber, related to mapping class group orbits.
problem Existence of finite orbits for higher Prym representations of the mapping class group.
method Study of surface-by-surface and surface-by-free groups.
result Existence of free-by-free and free-by-surface groups that do not algebraically fiber.
Poly-free groups are constructed as iterated semidirect products of free groups. The class of poly-free groups includes the classical pure braid groups, fundamental groups of fiber-type hyperplane arrangements, and certain subgroups of the automorphism groups of free groups. The purpose of this article is to compute ce…
Study shows hyperbolic subgroups can be free products of surface and free groups.
problem Characterizing hyperbolic subgroups within larger groups.
method Analyzing fiber bundles and using properties of hyperbolic groups.
result Non-elementary hyperbolic commensurated subgroups are virtually free products of surface and free groups.
Study knot invariants using automorphism groups of free nilpotent groups.
problem Developing knot invariants using automorphism groups.
method Nilpotently p-localization of knot groups and automorphism groups of free nilpotent groups. result Maps from outer automorphism groups yield knot invariants.
Free groups' automorphisms have bounded orbits.
problem Understanding automorphisms of infinite rank free groups.
method Proved coarsely bounded automorphism groups via metric space actions.
result Free groups' automorphism groups have quasi-isometry type of a point.
The study shows exponential distortion in virtually special groups containing free subgroups.
problem Understanding distortion in virtually special groups containing free subgroups.
method Constructing examples of virtually special groups with finite rank free subgroups.
result Distortion functions grow like exp^k(x^m) and can be superexponential.
Automorphisms of free groups yield invariant posets of lamination orbits.
problem Understanding the structure of free-by-cyclic groups through automorphisms.
method Analyzing the poset of attracting lamination orbits for free group automorphisms.
result The poset of lamination orbits is a commensurability invariant of free-by-cyclic groups.
The isomorphism problem for [free abelian]-by-free groups is unsolvable.
The study shows that surface groups are the only non-free infinite index subgroups of certain hyperbolic groups.
problem Identifying the only non-free infinite index subgroups of specific hyperbolic and one-relator groups.
method Careful analysis of free and cyclic splittings of cubulated groups.
result Proves that surface groups are the only non-free infinite index subgroups of certain hyperbolic and one-relator groups.
The paper extends Lipschitz metric isometries between Outer Spaces of virtually free groups.
problem Extending isometry properties of Lipschitz metric to virtually free groups.
method Analyzing finite-index subgroups and their covers, identifying folding paths, and using deformation retraction.
result Existence of candidates for Lipschitz distance and deformation retraction of spine.
We show, using Wise's equitable sets criterion, that every tubular free by cyclic group acts freely on a CAT(0) cube complex. We also show that these groups have a finite index subgroup satisfying the strongest Tits alternative, which means that every subgroup either surjects a non abelian free group or is torsion free…
The fundamental group of the complement of a hyperplane arrangement plays an important role in studying the corresponding arrangements. In particular, for large families of hyperplane arrangements, this fundamental group, being isomorphic to the fundamental group of a complement of a line arrangement, has some remarkab…
Proves a connectivity conjecture for free groups, showing homotopy type of spheres.
problem Establishing a connectivity conjecture for free groups.
method Provided homotopy-equivalent models of the common basis complex using free factors and sphere systems.
result The common basis complex of a free group of rank n has the homotopy type of a wedge of spheres of dimension 2n-3.
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.
The paper constructs representations of flat virtual braids by free group automorphisms.
problem Representing flat virtual braids by automorphisms of free groups.
method Construction of representations of flat virtual braid groups FVBn by automorphisms of free groups of rank 2n. result Established conditions of faithfulness and properties of the kernel for n≥3. We generalise the Karrass-Pietrowski-Solitar and the Nielsen realisation theorems from the setting of free groups to that of free products. As a result, we obtain a fixed point theorem for finite groups of outer automorphisms acting on the relative free splitting complex of Handel--Mosher and on the outer space of a fr…
An almost-direct product of free groups is an iterated semidirect product of finitely generated free groups in which the action of the constituent free groups on the homology of one another is trivial. We determine the structure of the cohomology ring of such a group. This is used to analyze the topological complexity …
The paper defines conditions for a free-by-free group to be hyperbolic.
problem Conditions for a free-by-free group to be relatively hyperbolic.
method Necessary and sufficient conditions involving exponentially growing automorphisms and invariant subgroup systems.
result A subgroup system can be used to construct peripheral subgroups making the extension hyperbolic.
We prove that the first order theory of nonabelian free groups eliminates the "there exists infinitely many" quantifier (in eq). Equivalently, since the theory of nonabelian free groups is stable, it does not have the finite cover property. We also extend our results to torsion-free hyperbolic groups under some conditi…
No hyperbolic group can have an infinite chain of free subgroups of fixed rank.
problem Infinite ascending chains of free subgroups in hyperbolic groups.
method Proof by contradiction and properties of hyperbolic groups.
result Hyperbolic groups do not contain strictly ascending chains of free quasiconvex subgroups of constant rank.
We give a simple proof of the finite presentation of Sela's limit groups by using free actions on R^n-trees. We first prove that Sela's limit groups do have a free action on an R^n-tree. We then prove that a finitely generated group having a free action on an R^n-tree can be obtained from free abelian groups and surfac…
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.
The paper classifies PD_4-complexes based on their fundamental group properties.
problem Understanding the structure of PD_4-complexes based on their fundamental group properties.
method Analyzing the fundamental group and its modules to classify PD_4-complexes.
result The classification of PD_4-complexes based on their fundamental group properties.
Conditions for hyperbolic and relatively hyperbolic extensions of free groups using automorphisms with fixed points.
problem Conditions for hyperbolic and relatively hyperbolic extensions of free groups.
method Using dynamics of outer automorphisms on the complex of free factors and investigating the geometry of the extension group.
result Conditions for hyperbolic and relatively hyperbolic extensions of free groups using automorphisms with fixed points.
Probabilistic proof shows separability in free groups.
problem Proving separability of infinite index subgroups in free groups.
method Probabilistic approach using homomorphisms onto alternating groups.
result Existence of a homomorphism preserving non-conjugacy.
We show that the complex of free factors of a free group of rank n > 1 is homotopy equivalent to a wedge of spheres of dimension n-2. We also prove that for n > 1, the complement of (unreduced) Outer space in the free splitting complex is homotopy equivalent to the complex of free factor systems and moreover is (n-2)-c…
We show that the natural map from the mapping class groups of surfaces to the automorphism groups of free groups, induces an infinite loop map on the classifying spaces of the stable groups after plus construction. The proof uses automorphisms of free groups with boundaries which play the role of mapping class groups o…
The group of 2-by-2 matrices with integer entries and determinant ±>1 can be identified either with the group of outer automorphisms of a rank two free group or with the group of isotopy classes of homeomorphisms of a 2-dimensional torus. Thus this group is the beginning of three natural sequences of groups, name…
We prove that the free splitting complex of a finite rank free group, also known as Hatcher's sphere complex, is hyperbolic.
Groups with similar pro-p completions have two-generated subgroups that are free.
problem Understanding the profinite rigidity of groups with specific pro-p completions. method Utilizing new results on residually-(torsion-free nilpotent) groups and their subgroups, and applying these to prove profinite rigidity.
result Groups with the same pro-p completion have two-generated subgroups that are free. We give a Dehn-Nielsen type theorem for the homology cobordism group of homology cylinders by considering its action on the acyclic closure, which was defined by Levine, of a free group. Then we construct an additive invariant of those homology cylinders which act on the acyclic closure trivially. We also describe some…
New method computes knot invariants using free group automorphisms.
problem Computing knot invariants efficiently and accurately.
method Using representations of braid groups by automorphisms of a free group.
result Compared isotopic invariants to Alexander polynomials.
Researchers show how to perturb free group representations into higher rank groups.
problem Understanding representations of free groups into non-Anosov Lie groups.
method Study quasi-isometric representations and show perturbations.
result Some representations can be perturbed to be non-quasi-isometric or unstable.