Study of uncountable family of finitely generated groups.
problem Characterize a family of finitely generated residually finite groups.
method Examined groups of the form F2∗HF2 where F2 is a rank-2 free group and H is an infinitely generated subgroup. result Unveiled uncountable family of groups up to isomorphism.
Finite groups generate only finitely many Kleinian groups.
problem Characterizing finite groups generating Kleinian groups.
method Generalized Gromov--Guth inequality and orbifold volume bounds.
result Only finitely many conjugacy classes of two-generated arithmetic Kleinian groups.
Finite type and finitely generated homotopy groups for manifold automorphisms.
problem Finite type and homotopy group properties of manifold automorphism spaces.
method Analyzing the classifying space of diffeomorphism groups and using simple homotopy theory.
result The classifying space of diffeomorphism groups has finitely generated homotopy groups.
The paper develops algorithms to detect stability and Morse properties in various groups.
problem Detecting stability and Morse properties in finitely generated groups.
method Various detection and decidability algorithms for stability and Morse properties in specific types of groups.
result The algorithms provide a way to determine if a subgroup is stable or Morse in specific group types.
A group fails a conjecture, constructed by researchers.
problem Generalized Burghelea Conjecture
method Constructed a specific group
result Failed to satisfy the generalized Burghelea conjecture
We show that every virtually torsion-free subgroup of the outer automorphism group of a conjugacy separable hyperbolic group is residually finite. As a result, we are able to prove that the group of outer automorphisms of every finitely generated Fuchsian group and of every free-by-finite group is resudually finite. We…
The paper characterizes Alexander quandles of finite groups.
problem Characterizing Alexander quandles of finite groups.
method Using group theory and automorphism groups, the paper provides characterizations of Alexander quandles.
result Generalized Alexander quandles of finite groups are characterized in terms of automorphism groups and underlying groups.
In geometric group theory one uses group actions on spaces to gain information about groups. One natural space to use is the Cayley graph of a group. The Cayley graph arguments that one encounters tend to require local finiteness, and hence finite generation of the group. In this paper, I take the theory of intersectio…
Proves finite generation of Johnson filtrations terms in mapping class groups and automorphism groups.
problem Finite generation of Johnson filtrations terms in mapping class groups and automorphism groups.
method Proves finite generation of terms in lower central series and Johnson filtrations of Torelli subgroups.
result Proves every term of the lower central series and Johnson filtrations of Torelli subgroups are finitely generated in a stable range.
New computations show various properties of bounded cohomology in finitely presented groups.
problem Understanding bounded cohomology properties in finitely presented groups.
method Computational and theoretical analysis of bounded cohomology.
result Existence of finitely presented non-amenable boundedly acyclic groups and groups with uncountable bounded cohomology.
The study proves how groups can be split with limited complexity.
problem Understanding the complexity of group splittings.
method Analyzing trees with finite stabilizers and their quotient structures.
result Deformation spaces of trees have maximal complexity.
A generalized Baumslag-Solitar (GBS) group is a finitely generated group acting on a tree with infinite cyclic edge and vertex stabilizers. We show how to determine effectively the rank (minimal cardinality of a generating set) of a GBS group; as a consequence, one can compute the rank of the mapping torus of a finite …
We construct analogues of FI-modules where the role of the symmetric group is played by the general linear groups and the symplectic groups over finite rings and prove basic structural properties such as Noetherianity. Applications include a proof of the Lannes--Schwartz Artinian conjecture in the generic representatio…
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…
Groups with semistable peripheral subgroups are semistable.
problem Semistability of fundamental groups in relatively hyperbolic groups.
method Generalization of semistability from 1-ended subgroups to finitely generated subgroups with semistable fundamental groups.
result Semistability of fundamental groups in more general relatively hyperbolic groups.
Johnson filtrations of mapping class groups are finitely generated.
problem Finiteness of Johnson filtrations in mapping class groups.
method Proved linear stable range for lower central series and Johnson filtrations of Torelli subgroups.
result Every term of the filtrations is finitely generated.
3-manifold groups are uniquely identifiable via their profinite completions.
problem Identifying uniquely 3-manifold groups among finitely generated groups.
method Proving that the inclusion of profinite completions is not an isomorphism for proper subgroups.
result All finitely generated 3-manifold groups are Grothendieck rigid.
We describe a procedure for constructing a generalized Thompson group out of a family of groups that is equipped with what we call a cloning system. The previously known Thompson groups F, V, Vbr and Fbr arise from this procedure using, respectively, the systems of trivial groups, symmetric groups, braid groups and pur…
Torelli group's second homology is finite for large genus surfaces.
problem Determining the finiteness of Torelli group's homology.
method Analyzing rational homology groups of Torelli groups for surfaces of genus g ≥ 51.
result Second rational homology of Torelli group is finite for large genus surfaces.
We give an exposition of Delzant's ideas extending the notion of Scott complexity of finitely generated groups to surjective homomorphisms of finitely presented groups to finitely generated groups.
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 …
Paper proves Torelli group's finiteness for surfaces with 2 boundaries.
problem Finiteness of Torelli group for surfaces with 2 boundaries.
method Analyzes Torelli group properties for surfaces of genus ≥ 3 with 2 boundary components.
result Torelli group is finitely generated for surfaces of genus ≥ 3 with 2 boundaries.
Triangle groups uniquely identified by their finite quotients.
problem Identifying triangle groups among finitely generated residually finite groups.
method Character varieties method to distinguish profinite completions.
result Certain Fuchsian triangle groups are profinitely rigid.
Study covers of Chamanara surface with large Veech groups.
problem Characterize finite abelian covers with large Veech groups.
method Analyze Veech groups of finite abelian covers of the Chamanara surface.
result Every free group arises as the projective Veech group of a finite-area infinite translation surface.
In this paper we provide a criteria for geometric finiteness of Kleinian groups in general dimension. We formulate the concept of conformal finiteness for Kleinian groups in space of dimension higher than two, which generalizes the notion of analytic finiteness in dimension two. Then we extend the argument in the paper…
Finite index subgroups of relatively hyperbolic groups have equal index.
problem Finite index subgroups of relatively hyperbolic groups have equal index.
method Demonstrating that the number of simplices in a simplicial classifying space grows linearly with index.
result Finite index subgroups of relatively hyperbolic groups have equal index.
Many 2D Artin groups are residually finite.
problem Residual finiteness of 2D Artin groups.
method Showed these groups split as free products with amalgamation or HNN extensions of finite rank free groups.
result Many 2D Artin groups are residually finite.
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.
The study examines Hopfian properties of conjugation quandles and their underlying groups.
problem Understanding the relationship between Hopfian properties of conjugation quandles and their underlying groups.
method Examined Hopfian and residual finiteness properties of conjugation quandles of specific groups.
result Conjugation quandles of Baumslag-Solitar groups are infinitely generated and not necessarily Hopfian.
The paper computes presentations of cluster modular groups and verifies their generation by Dehn twists.
problem Computing presentations and verifying generation of cluster modular groups.
method A method to compute presentations of saturated cluster modular groups and verification of generation by cluster Dehn twists.
result The cluster modular groups of specified types are virtually generated by cluster Dehn twists.
The paper establishes a correspondence between orbit braid groups and a finite generated group.
problem Generalizing Artin's ideas to establish a correspondence between orbit braid groups and a finite generated group.
method Finding a faithful representation of the orbit braid group in a finite generated group and investigating characterizations of the orbit braid representation.
result A one-to-one correspondence between the orbit braid group and a quotient of a group formed by homeomorphisms of a punctured plane.
Abstract: Presents finite subgroups of genus 2 surface mapping class group.
problem Identifying all finite subgroups of the mapping class group of a genus 2 surface.
method Using Humphries generators to present finite subgroups of the mapping class group of a genus 2 surface.
result Presented presentations of all finite subgroups of the mapping class group of a closed surface of genus 2.
Study shows intersections of stable subgroups have finite properties.
problem Understanding intersections of stable subgroups in finitely generated groups.
method Analyzing intersections of conjugates and extending quasimorphisms.
result Finite families of quasimorphisms can be simultaneously extended to the ambient group.
Regular languages describe contracting geodesics in groups.
problem Characterizing groups with infinite contracting geodesics.
method Analyzing geodesics in Cayley graphs with a contracting property.
result Groups with infinite contracting geodesics are either virtually Z or acylindrically hyperbolic.
Flawed groups are shown to include all finitely generated groups isomorphic to free products of nilpotent groups.
problem Characterizing flawed groups and understanding their topological properties.
method Analyzing finitely presented groups and their deformation retracts onto subspaces of character varieties.
result All finitely generated groups isomorphic to free products of nilpotent groups are flawed.
New groups without finite classifying spaces found in Kodaira fibrations.
problem Finding groups without finite classifying spaces.
method Constructing Kähler groups as fundamental groups of Kodaira fibrations.
result Found Kähler groups that are not commensurable to surface groups and do not have finite classifying spaces.
Genus 3 Heegaard groups of lens space connected sums are finitely generated.
problem Understanding the structure of mapping class groups of genus 3 Heegaard splittings.
method Proved finitely generated property through connected reducing sphere complexes.
result Mapping class groups are finitely generated and complexes are connected.
Study SL(2,C) character schemes for finitely generated groups.
problem Characterize SL(2,C) representations of finitely generated groups.
method Define coordinate rings and equations for SL(2,C) character schemes.
result Explicit equations for character schemes of finitely presented groups.
We show that every graph product of finitely generated abelian groups acts properly and cocompactly on a CAT(0) cubical complex. The complex generalizes (up to subdivision) the Salvetti complex of a right-angled Artin group and the Coxeter complex of a right-angled Coxeter group. In the right-angled Artin group case it…
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. This thesis introduces big mapping class groups and their structure.
problem Understanding mapping class groups of infinite-type surfaces.
method Systematic introduction and analysis of structure and topological generation.
result Key differences from finite-type mapping class groups.
We survey the existing parts of a classification of finite groups generated by orthogonal transformations in a finite-dimensional Euclidean space whose fixed point subspace has codimension one or two and extend it to a complete classification. These groups naturally arise in the study of the quotient of a Euclidean spa…
Study on commutator subgroups of welded braid groups, proving their finiteness and perfection.
problem Investigating the structure of commutator subgroups in welded braid groups.
method Proved finiteness and Hopfian property, showed perfection for n≥5, computed finite presentations. result Commutator subgroups of welded braid groups are finitely generated, Hopfian, and perfect for n≥5. Paper shows 3D hyperbolic manifolds are uniquely identified by their finite groups.
problem Identifying hyperbolic 3-manifolds using their finite quotient groups.
method Proves profinite isomorphism of fundamental groups uniquely determines the manifold's isomorphism type.
result Finite-volume hyperbolic 3-manifolds are almost determined by their finite quotient groups.
Extensions of Veech groups using hierarchical hyperbolic spaces.
problem Generalizing Veech groups to non-lattice cases.
method Using a hyperbolic space E^ and a nice action of Γ on it. result Finitely generated Veech groups are hierarchically hyperbolic.
Study of quasiconvex subgroups in 3-manifold groups.
problem Characterize quasiconvex subgroups in 3-manifold groups.
method Analyzes strongly quasiconvex subgroups in finitely generated 3-manifold groups.
result Characterizes quasiconvex subgroups in graph manifold groups and 3-manifold groups.
Study of generalized J-groups and their presentations.
problem Understanding the structure of generalized J-groups and their presentations.
method Determine finitely generated groups, classify up to reflection isomorphism, and derive explicit presentations.
result Generalized J-groups coincide with rank 2 complex reflection groups and their torsion quotients.
The paper studies groups with specific actions on hyperbolic spaces and finds that subgroups are either amenable or contain a free group.
problem Characterizing subgroups of groups with specific actions on hyperbolic spaces.
method Analyzing groups with property (PPH) and (PPT) and their subgroups.
result Any finitely generated subgroup of a finitely generated group with property (PPH) either is amenable or contains \(F_2\).