arXiv research

Characterizes a general range decreasing group homomorphism.

We extend certain homomorphisms defined on the higher Torelli subgroups of the mapping class group to crossed homomorphisms defined on the entire mapping class group. In particular, for every $k\geq 2$, we construct a crossed homomorphism $ε_k$ which extends Morita's homomorphism $\tilde τ_k$ to the entire mapping clas…

Two crossing homomorphisms on braid groups are shown to be equivalent.

Study homomorphisms from groups to 3-manifold fundamental groups.

The study classifies homomorphisms from mapping class groups using finite subgroups.

Stability of Lie group homomorphisms and subgroups via Moser type argument.

Classifies homomorphisms between specific braid groups.

New conditions for weighted composition operators in group homomorphisms.

Homomorphism from braid groups to Steinberg groups defined.

Classifies homomorphisms from braid groups, proving their extensions to automorphisms.

The paper extends Johnson homomorphisms to groups with extended N-series.

A new homomorphism connects group actions on circles to Euler classes.

We extend each higher Johnson homomorphism to a crossed homomorphism from the automorphism group of a finite-rank free group to a finite-rank abelian group. We also extend each Morita homomorphism to a crossed homomorphism from the mapping class group of once-bounded surface to a finite-rank abelian group. This improve…

Study classifies biharmonic and harmonic homomorphisms between specific Lie groups.

Study proves equality of LS-category and cohomological dimension for specific group homomorphisms.

Study of handlebody group intersections with Torelli and Johnson kernels.

Paper classifies totally symmetric sets in groups and bounds their sizes.

New quasi-homomorphisms on surface groups vanish on handlebody groups.

The paper shows how surjective homomorphisms between surface braid groups factor and computes their automorphism groups.

Johnson and Livingston have characterized peripheral structures in homomorphs of knot groups. We extend their approach to the case of links. The main result is an algebraic characterization of all possible peripheral structures in certain homomorphic images of link groups.

Let K be the kernel of an epimorphism G -> Z, where G is a finitely presented group. If K has infinitely many subgroups of index 2, 3, or 4, then it has uncountably many. Moreover, if K is the commutator subgroup of a classical knot group G, then any homomorphism from K onto the symmetric group S_2 lifts to a homomorph…

The paper studies symmetries in quandles and their relative versions.

Classifies homomorphisms from braid groups to mapping class groups of nonorientable surfaces.

We introduce the notion of tight homomorphism into a locally compact group with nonvanishing bounded cohomology and study these homomorphisms in detail when the target is a Lie group of Hermitian type. Tight homomorphisms between Lie groups of Hermitian type give rise to tight totally geodesic maps of Hermitian symmetr…

Periodic links' groups relate via homomorphisms to finite groups.

Let $M$ and $N$ be two closed $C^{\infty}$ manifolds and let $\text{Diff}_c(M)$ denote the group of $C^{\infty}$ diffeomorphisms isotopic to the identity. We prove that any (discrete) group homomorphism between $\text{Diff}_c(M)$ and $\text{Diff}_c(N)$ is continuous. We also show that a non-trivial group homomorphism $…

Homomorphisms between pure mapping class groups are classified for certain genus surfaces.

This paper concerns rigidity of the mapping class groups. We show that any homomorphism $φ:{\rm Mod}_g\to {\rm Mod}_h$ between mapping class groups of closed orientable surfaces with distinct genera $g>h$ is trivial if $g\geq 3$ and has finite image for all $g\geq 1$. Some implications are drawn for more general homomo…

Study on totally symmetric sets with group applications.

If phi: G-->G' is a surjective homomorphism, we prove that the twisted Alexander polynomial of G is divisible by the twisted Alexander polynomial of G'. As an application, we show non-existence of surjective homomorphism between certain knot groups.

The paper classifies homomorphisms between braid groups and mapping class groups.

We prove that either the images of the mapping class groups by quantum representations are not isomorphic to higher rank lattices or else the kernels have a large number of normal generators. Further we show that the images of the mapping class groups have nontrivial 2-cohomology, at least for small levels. For this pu…

We propose an approach to study non-Abelian Iwasawa theory, using the idea of Johnson homomorphisms in low dimensional topology. We introduce arithmetic analogues of Johnson homomorphisms/maps, called the p-Johnson homomorphisms/maps, associated to the Zassenhaus filtration of a pro-p Galois group over a Z_p-extension …

Researchers determine the rational abelianization of a subgroup of mapping class groups.

We refine the construction of quasi-homomorphisms on mapping class groups. It is useful to know that there are unbounded quasi-homomorphisms which are bounded when restricted to particular subgroups since then one deduces that the mapping class group is not boundedly generated by these subgroups. In this note we enlarg…

Satellite operations with winding number ≠ 1 are not homomorphisms.

This is partly a survey and partly a research article. Some known results and open problems about Kaehler groups (fundamental groups of compact Kaehler manifolds) are discussed. A new notion of Kaehler homomorphism is introduced. This is a homomorphism induced by a holomorphic map between these types of manifolds. Some…

Study on flux homomorphism and its extension in symplectic group of a disk.

Johnson has defined a surjective homomorphism from the Torelli subgroup of the mapping class group of the surface of genus $g$ with one boundary component to $\wedge^3 H$, the third exterior product of the homology of the surface. Morita then extended Johnson's homomorphism to a homomorphism from the entire mapping cla…

The paper extends Johnson's result on Torelli group homology.

The paper studies new filtrations and homomorphisms related to mapping class groups and 3-manifold invariants.

New proof shows homomorphisms from pure braid groups to hyperbolic groups have cyclic images or factor through forgetful maps.

Kawauchi defined a group structure on the set of homology $S^1$\times$S^2$'s under an equivalence relation called $\widetilde{H}$-cobordism. This group receives a homomorphism from the knot concordance group, given by the operation of zero-surgery. It is natural to ask whether the zero-surgery homomorphism is injecti…

The paper studies trace-preserving homomorphisms on SL(2,C) and their implications.

Paper defines and analyzes Chillingworth classes for subsurface Torelli groups.

We give a new proof of a celebrated theorem of Dennis Johnson that asserts that the kernel of the Johnson homomorphism on the Torelli subgroup of the mapping class group is generated by separating twists. In fact, we prove a more general result that also applies to "subsurface Torelli groups". Using this, we extend Joh…

Complexity of homomorphisms from 3-manifolds to simple groups is #P-complete.

We give elementary applications of quasi-homomorphisms to growth problems in groups. A particular case concerns the number of torsion elements required to factorise a given element in the mapping class group of a surface.