The paper explores geometric finiteness in mapping class groups and constructs new examples of these subgroups.
problem Understanding geometric finiteness in mapping class groups and constructing new examples.
method Examined several constructions of subgroups and determined conditions for geometric finiteness.
result Provides new examples of parabolically geometrically finite and reducibly geometrically finite subgroups.
Paper explores hyperbolic groups with unique subgroup properties.
problem Creating examples of hyperbolic groups with specific subgroup finiteness properties.
method Develops new examples of subgroups of hyperbolic groups with various finiteness properties.
result Uncountably many groups of type FP2 with similar properties to subgroups of hyperbolic groups. 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.
The paper disproves the existence of certain subgroups with nontrivial rational abelianization.
problem The existence of finite-index subgroups with nontrivial rational abelianization in handlebody groups.
method Proved that meridian multitwists vanish in H1(Γ;Q) and showed that H1(Γ;Q)=0 for specific subgroups. result No finite-index subgroups of the handlebody group contain nontrivial rational abelianization.
The study characterizes subgroups of braid groups and their finite quotients.
problem Understanding the relationship between braid groups and their congruence subgroups.
method Characterization and computation of finite quotients of braid groups.
result Explicit generators and free subgroups are found for certain braid groups.
We show that for certain arithmetic groups, geometrically finite subgroups are the intersection of finite index subgroups containing them. Examples are the Bianchi groups and the Seifert-Weber dodecahedral space. In particular, for manifolds commensurable with these groups, immersed incompressible surfaces lift to embe…
Suppose that all hyperbolic groups are residually finite. The following statements follow: In relatively hyperbolic groups with peripheral structures consisting of finitely generated nilpotent subgroups, quasiconvex subgroups are separable; Geometrically finite subgroups of non-uniform lattices in rank one symmetric sp…
The paper disproves a conjecture about isomorphic subgroups in finite groups.
problem The existence of non-isomorphic subgroups that are isomorphic in extensions of finite groups.
method Constructing extensions of finite groups to show non-isomorphic pre-images of subgroups.
result Subgroups of finite groups that are isomorphic in extensions are not conjugate.
Nearly Fuchsian subgroups found in certain Kleinian groups.
problem Finding surface subgroups in finite covolume Kleinian groups.
method Proving K-quasiconformal conjugacy to Fuchsian groups.
result Every finite covolume Kleinian group has a nearly Fuchsian surface subgroup.
Characterizes groups arising as fixed subgroups of RAAG automorphisms.
problem Identifying groups that can be fixed by finite-order automorphisms of RAAGs.
method Geometric characterisation using divisible cube complexes.
result Surface groups and commutator subgroups of RAAGs are fixed subgroups.
Example shows dense subgroup of SL5(Z) not finitely presented.
problem Finding dense subgroups of SL5(Z) that are not finitely presented.
method Discussing an example of a Zariski-dense finitely generated subgroup of SL5(Z).
result Example shows a subgroup that is dense but not finitely presented.
We show that a finite collection of stable subgroups of a finitely generated group has finite height, finite width and bounded packing. We then use knowledge about intersections of conjugates to characterize finite families of quasimorphisms on hyperbolically embedded subgroups that can be to simultaneously extended to…
The study examines discrete subgroups of Lie groups and their residual finiteness.
problem Determining when discrete subgroups of Lie groups are residually finite.
method Analyzes known results and poses open questions.
result Answers to open questions will provide a comprehensive understanding of residual finiteness in Lie groups.
The paper calculates subgroup distortions in 3-manifold groups.
problem Understanding subgroup distortions in 3-manifold groups.
method Computed all finitely generated subgroups of finitely generated 3-manifold groups and analyzed their distortions.
result Subgroup distortions in 3-manifold groups are linear, quadratic, exponential, or double exponential.
Proves congruence subgroup property for mapping class groups of hyperbolic surfaces.
problem Residual finiteness of hyperbolic groups and congruence subgroup property for mapping class groups.
method Assumption of residual finiteness of hyperbolic groups leads to proof of congruence subgroup property.
result Congruence subgroup property for mapping class groups of hyperbolic surfaces.
We say A is a quasi-normal subgroup of the group G if the commensurator of A in G is all of G. We develop geometric versions of commensurators in finitely generated groups. In particular, g is an element of the commensurator of A in G iff the Hausdorff distance between A and gA is finite. We show that a quasi-normal su…
Finite groups of RAAGs act on a contractible space without fixed points.
problem Characterizing finite subgroups of RAAG outer automorphisms.
method Analyzing the action of finite subgroups on a contractible space.
result Finite subgroups of RAAG outer automorphisms fix a point in the space.
Finite subgroups of good groups correspond to their profinite completions.
problem Characterizing finite subgroups of profinite completions of good groups.
method Proving bijective correspondence between conjugacy classes of finite p-subgroups in G and G^, and analyzing centralizers and normalizers. result Bijective correspondence between conjugacy classes of finite p-subgroups in G and G^. Finite p-group actions on manifolds have limited stabilizer subgroups.
problem Understanding the structure of stabilizer subgroups in group actions on manifolds.
method Bounding the index of a subgroup H in a finite p-group G acting on a compact manifold M, ensuring a controlled number of stabilizers.
result The existence of a subgroup H with a controlled index and limited stabilizers.
Researchers find a way to bound the complexity of certain subgroup geometric invariants.
problem Understanding the geometric invariants of subgroups of direct products of free groups.
method Generalizing techniques for 'pushing fillings' into normal subgroups.
result Finitely presented subgroups of direct products of three free groups and subgroups of finiteness type Fn−1 in a direct product of n free groups have Dehn functions bounded above by N9. Study complex hyperbolic lattices and their subgroups, proving new finiteness properties.
problem Characterize subgroups of complex hyperbolic lattices.
method Analyzing homomorphisms and using arithmetic lattice properties.
result Deep subgroups of complex hyperbolic lattices admit homomorphisms to Z with specific kernel types.
The study finds a limit on subgroup complexity in hyperbolic 3-manifold groups.
problem Understanding subgroups of bounded rank in hyperbolic 3-manifold groups.
method Proving a finiteness theorem for subgroups of bounded rank.
result Every bounded rank covering tower of closed hyperbolic 3-manifolds is a tower of finite covers associated to a fibration over a 1-orbifold.
If F is a surface with boundary, then a finitely generated subgroup without peripheral elements of G = π_1(F) can be separated from finitely many other elements of G by a finite index subgroup of G corresponding to a finite cover F' with the same number of boundary components as F .
Closed Lorentz 4-manifolds have finite isometry groups with a bounded abelian subgroup.
problem Understanding the structure of isometry groups of closed Lorentz 4-manifolds.
method Proving that any finite subgroup of the isometry group of a closed Lorentz 4-manifold has a bounded abelian subgroup.
result Finite isometry groups of closed Lorentz 4-manifolds have a bounded abelian subgroup.
The main result of this article is a refinement of the well-known subgroup separability results of Hall and Scott for free and surface groups. We show that for any finitely generated subgroup, there is a finite dimensional representation of the free or surface group that separates the subgroup in the induced Zariski to…
We show that if a group is not virtually cyclic and is hyperbolic relative to a family of proper subgroups, then it has a hyperbolically embedded subgroup which contains a finitely generated non-abelian free group as a finite index subgroup.
Finite index subgroups of certain groups cannot act faithfully on the circle.
problem Finite index subgroups of specific groups cannot act faithfully on the circle.
method Analyzing C1 actions on the circle for finite index subgroups of mapping class groups, automorphism groups, and outer automorphism groups. result No orientation preserving C1 action of finite index subgroups of these groups on the circle can be faithful. Let G be a lattice in PSL(2,C). The pro-normal topology on G is defined by taking all cosets of non-trivial normal subgroups as a basis. This topology is finer than the pro-finite topology, but it is not discrete. We prove that every finitely generated subgroup H<G is closed in the pro-normal topology. As a corollary w…
For a finitely generated group, there are two recent generalizations of the notion of a quasiconvex subgroup of a word-hyperbolic group, namely a stable subgroup and a Morse or strongly quasiconvex subgroup. Durham and Taylor defined stability and proved stability is equivalent to convex cocompactness in mapping class …
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.
The theorems of M. Ratner, describing the finite ergodic invariant measures and the orbit closures for unipotent flows on homogeneous spaces of Lie groups, are extended for actions of subgroups generated by unipotent elements. More precisely: Let G be a Lie group (not necessarily connected) and Gamma a closed subgroup …
It is known that the level 2 principal congruence subgroup of GL(n;Z) has a finite generating set. In this paper, we give a finite presentation of the level 2 principal congruence subgroup of GL(n;Z).
Study of subgroups in complex hyperbolic lattice triangle groups.
problem Characterizing subgroups of finite index in complex hyperbolic lattice triangle groups.
method Explicit construction and analysis of subgroups, examination of their properties.
result Identification of neat subgroups, subgroups with positive first Betti number, and homomorphisms onto non-Abelian free groups.
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.
Study ramification in knot groups through finite covers and their quotients.
problem Understanding ramification in knot groups and their covers.
method Formalized ramification theory for knot groups, analyzed through finite quotients, profinite completions, and cohomology.
result Characterized ramification and inertia subgroups in knot groups and their covers.
A dense amalgam connects boundaries of groups split by finite subgroups.
problem Understanding boundaries of groups split by finite subgroups.
method Introducing dense amalgam and applying it to EZ-boundaries. result Boundaries of groups split by finite subgroups have a dense amalgam structure.
A celebrated theorem of Marshall Hall Jr. implies that finitely generated free groups are subgroup separable and that all of their finitely generated subgroups are retracts of finite-index subgroups. We use topological techniques inspired by the work of Stallings to prove that all limit groups share these two propertie…
The orbifold group of the Borromean rings with singular angle 90 degrees, U, is a universal group, because every closed oriented 3--manifold M3 occurs as a quotient space M3=H3/G, where G is a finite index subgroup of U. Therefore, an interesting, but quite difficult problem, is to classify the fin…
Proofs show finite subgroups of homeomorphism groups are almost nilpotent.
problem Finite subgroups of homeomorphism groups of compact topological manifolds.
method Finite group theoretic results provide a general strategy for proving Jordan-type theorems.
result Proof of the revised Ghys conjecture about nilpotent normal subgroups.
Consider a group G and a family A of subgroups of G. We say that vertex finiteness holds for splittings of G over A if, up to isomorphism, there are only finitely many possibilities for vertex stabilizers of minimal G-trees with edge stabilizers in A. We show vertex finiteness when G…
Study subgroups of pro-p PD^3 groups, finding specific conditions.
problem Characterize subgroups of pro-p PD^3 groups. method Analyzes properties of subnormal and finitely presented subgroups.
result Conditions on subgroups of pro-p PD^3 groups. 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.
Unified algebraic framework for virtual braid structures with strong structural consequences.
problem Unified algebraic framework for virtual braid structures with various types of crossings.
method Introducing the universal virtual braid group UVn(c) and proving its properties. result Strong structural consequences including residual finiteness, linearity, and solvability of conjugacy problems.
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 …
Let Mod(S) be the extended mapping class group of a surface S. For S the twice-punctured torus, we show that there exists an isomorphism of finite index subgroups of Mod(S) which is not the restriction of an inner automorphism. For S a torus with at least three punctures, we show that every injection of a finite index …
Torelli subgroup rigidity in Out(F_N) proven for N≥4.
problem Proving rigidity of Torelli subgroup in Out(F_N).
method Injective homomorphisms and conjugation analysis.
result Every injective homomorphism from Torelli subgroup to Out(F_N) is conjugate to inclusion.
We prove that a dense subgroup of Homeo+(I) is not elementary amenable. We also show that the topological group Homeo+(I) does not satisfy the Stability of the Generators Property, moreover, any finitely generated subgroup of Homeo+(I) admits a faithful discrete representation …
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.