New concept SB-generation helps classify transformation groups.
problem Classifying transformation groups through quasi-isometry invariants.
method Identifying SB-generated groups in specific transformation groups.
result SB-generation provides robust extension of finite generation.
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.
Proves a minimal generating set for a specific group of mapping classes.
problem Finding a minimal generating set for a specific group of mapping classes.
method Proved the group is generated by four elements, with minimal exceptions.
result Minimal generating set for the balanced superelliptic mapping class group.
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.
The notions of stable and Morse subgroups of finitely generated groups generalize the concept of a quasiconvex subgroup of a word-hyperbolic group. For a word-hyperbolic group G, Kapovich provided a partial algorithm which, on input a finite set S of G, halts if S generates a quasiconvex subgroup of G and run…
A generalized Baumslag-Solitar group is the fundamental group of a graph of groups all of whose vertex and edge groups are infinite cyclic. Levitt proves that any generalized Baumslag-Solitar group has property R-infinity, that is, any automorphism has an infinite number of twisted conjugacy classes. We show that any g…
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…
Generic groups satisfy a chain condition for subgroups.
problem Understanding subgroup structures in generic groups.
method Proving for fixed integers m,t,k in generic m-generator t-relator groups. result Generic groups satisfy the Ascending Chain Condition for k-generated subgroups. 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.
Paper solves isomorphism problem for specific Baumslag-Solitar groups.
problem Isomorphism problem for small rose non-ascending generalized Baumslag-Solitar groups.
method Analyzed group actions on trees with specific stabilizers.
result Isomorphism problem solvable for the specified groups.
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…
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.
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 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.
We ask if any finite type generalized braid group is a subgroup of some classical Artin braid group. We define a natural map from a given finite type generalized braid group to a classical braid group and ask if this map is an injective homomorphism. We prove that this map is a homomorphism for the braid groups of type…
A group, defined as set with associative multiplication and inverse, is a natural structure describing the symmetry of a space. The concept of group generalizes to group objects internal to other categories than sets. But there are yet more general objects that can still be thought of as groups in many ways, such as qu…
The study restricts normal subgroups of Kähler groups, proving specific cases and general restrictions.
problem Characterizing normal subgroups of Kähler groups.
method Analyzing embeddings and conjugation actions of surface groups and one-ended hyperbolic groups.
result Restrictions on normal subgroups of Kähler groups, including virtual direct products and surface group properties.
Three elements generate balanced superelliptic mapping class groups.
problem Generating balanced superelliptic mapping class groups.
method Proving groups are generated by three elements through normalizers and liftable mapping class groups.
result Balanced superelliptic mapping class groups are generated by three elements.
Study shows Morse elements are common in acylindrically hyperbolic groups.
problem Understanding generic elements in acylindrically hyperbolic groups.
method Analyzing Morse elements and outer automorphisms.
result Morse elements are common in acylindrically hyperbolic groups.
Paper proves vanishing homology groups for certain hyperbolic groups.
problem Understanding homology groups of specific hyperbolic groups.
method Using twisted Wirtinger presentations to prove homology group vanishing.
result Second homology groups vanish for certain Gromov hyperbolic groups.
Minimal generating sets found for Kim-Manturov groups.
problem Understanding the structure of groups related to surface triangulations.
method Provided minimal generating sets and determined abelianizations.
result Minimal generating sets and abelianization results for the groups.
We classify connected Lie groups which are locally isomorphic to generalized Heisenberg groups. For a given generalized Heisenberg group N, there is a one-to-one correspondence between the set of isomorphism classes of connected Lie groups which are locally isomorphic to N and a union of certain quotients of noncom…
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.
We classify the groups quasi-isometric to a group generated by finite-order elements within the class of one-ended hyperbolic groups which are not Fuchsian and whose JSJ decomposition over two-ended subgroups does not contain rigid vertex groups. To do this, we characterize which JSJ trees of a group in this class admi…
New method shows any triangle group generating pair is related to special coverings.
problem Understanding generating pairs of triangle groups.
method Special almost orbifold coverings.
result Any generating pair of a triangle group is represented by a special covering.
Study homomorphisms from groups to 3-manifold fundamental groups.
problem Understanding homomorphisms between groups and 3-manifold fundamental groups.
method Proves foundational results and answers specific questions.
result Answers questions posed by Reid-Wang-Zhou and Agol-Liu.
The class of acylindrically hyperbolic groups, which are groups that admit a certain type of non-elementary action on a hyperbolic space, contains many interesting groups such as non-exceptional mapping class groups and Out(Fn) for n≥2. In such a group, a generalized loxodromic element i…
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 groups can't be fundamental groups of symplectic Calabi-Yau manifolds.
problem Proving certain groups cannot be fundamental groups of symplectic Calabi-Yau manifolds.
method Using the method of Friedl and Vidussi, they showed these groups cannot be fundamental groups of SCY manifolds.
result The Lodha--Moore groups and their n-adic generalizations cannot be fundamental groups of any SCY manifold. The study of double coset growth in specific groups confirms a conjecture about generic 3-manifolds.
problem Understanding the growth of double cosets in specific groups.
method Generalizing previous work on hyperbolic groups, the study proves the growth of double cosets for certain subgroups.
result The double coset growth of certain subgroups is comparable to the orbital growth function.
The Gehring-Martin-Tan inequality for 2-generator subgroups of PSL(2,C) is one of the best known discreteness conditions. A Kleinian group G is called a Gehring-Martin-Tan group if the equality holds for the group G. We give a method for constructing Gehring-Martin-Tan groups with a generator of order four and pres…
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.
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. 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.
New Lie groups generalize H-type groups with nondegenerate centers.
problem Generalizing H-type groups with nondegenerate centers.
method Defined and investigated 2-step nilpotent Lie groups.
result Geometric properties of new Lie groups investigated.
Study stabilizers of complex hyperbolic triangle groups, finding generators and signatures.
problem Understanding the stabilizers of complex hyperbolic triangle groups.
method Explicit generators and signatures of stabilizers computed for each group orbit of mirrors.
result Explicit generators and signatures of stabilizers for some triangle groups.
The paper finds minimal generating sets and abelianizes the quasitoric braid group.
problem Understanding the structure of quasitoric braids and their subgroup properties.
method Provided two minimal generating sets and determined the abelianization.
result Minimal generating sets and abelianization of the quasitoric braid group were determined.
A nontrivial element in a group is a generalized torsion element if some nonempty finite product of its conjugates is the identity. We prove that any generalized torsion element in a free product of torsion-free groups is conjugate to a generalized torsion element in some factor group. This implies that the fundamental…
Introduces Θ-positivity in Lie groups, generalizing Lusztig's positivity.
problem Generalizing Lusztig's total positivity to a broader class of Lie groups.
method Introduces and studies Θ-positivity in real simple Lie groups. result Four families of Lie groups admit Θ-positive structures. 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…
Study shows quotient group is infinitely generated.
problem Understanding the structure of homology cobordism groups.
method Proved using Seifert fibered spaces.
result Quotient group is infinitely generated.
We prove twisted homological stability with polynomial coefficients for automorphism groups of free nilpotent groups of any given class. These groups interpolate between two extremes for which homological stability was known before, the general linear groups over the integers and the automorphism groups of free groups.…
Factorizes discrete representations of finitely generated groups into PSL(2, R).
problem Understanding discrete representations of finitely generated groups into PSL(2, R).
method Factorization theorem for Fuchsian groups, Makanin-Razborov diagrams, and new class of groups called PSL(2, R)-discrete limit groups.
result Obtained useful information about PSL(2, R)-discrete limit groups.
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.
The study finds abundant normal generators for mapping class groups.
problem Understanding normal generation in mapping class groups.
method Analyzing restrictions on invariant subsurfaces and Teichmüller spaces.
result Reducible mapping classes can normally generate mapping class groups based on their asymptotic translation lengths.
New examples of Howson groups that are not strongly Howson found.
problem Understanding the difference between Howson and strongly Howson groups.
method Constructing specific examples of groups to demonstrate the distinction.
result First examples of Howson groups that are not strongly Howson.
We determine the number of connected components of the moduli space for representations of a surface group in the general linear group.
This book offers to study locally compact groups from the point of view of appropriate metrics that can be defined on them, in other words to study "Infinite groups as geometric objects", as Gromov writes it in the title of a famous article. The theme has often been restricted to finitely generated groups, but it can f…