arXiv research

Absolutely partially hyperbolic surface endomorphisms have a coherent center foliation.

The paper classifies fixed subgroups of endomorphisms in free-abelian times surface groups.

We prove that every endomorphism of the mapping class group of an orientable surface onto a subgroup of finite index is in fact an automorphism.

We address the problem of necessary conditions and topological obstructions for the existence of robustly transitive maps on surfaces. Concretely, we show that partial hyperbolicity is a necessary condition in order to have $C^1$ robustly transitive endomorphisms with critical points on surfaces, and the only surfaces …

For a closed surface $S$ with $χ(S)<0$, we show that the fixed subgroup of a family $\mathcal B$ of endomorphisms of $π_1(S)$ has $\rk \fix\mathcal B\leq \rk π_1(S)$. In particular, if $\mathcal B$ contains a non-epimorphic endomorphism, then $\rk \fix\mathcal B\leq \frac{1}{2} \rk π_1(S)$. We also show that geometric …

In-degree quiver polynomials for surface-links computed.

This paper exhausts curve complexes on non-orientable surfaces.

This paper studies fixed points of graph selfmaps and iwip endomorphisms of free groups.

The paper extends the Manhattan curve concept to complex dynamics and studies its relation to multiplier spectra.

The instanton Floer homology of a knot in the three-sphere is a vector space with a canonical mod 2 grading. It carries a distinguished endomorphism of even degree,arising from the 2-dimensional homology class represented by a Seifert surface. The Floer homology decomposes as a direct sum of the generalized eigenspaces…

Study fixed point indices and words at infinity for graph selfmaps.

New surfaces in Lorentz-Minkowski space with constant mean curvature identified.

We show that all non-trivial continuous endomorphisms of the circle group are topologically mixing. We also show that there exists a large infinite class of continuous endomorphisms of any n-dimensional torus group which are topologically mixing. Lastly, we prove that any continuous endomorphism on an abelian polish se…

Quotients of torus endomorphisms have parabolic orbifolds.

The paper defines constraints for commuting endomorphisms in generalized tangent bundles.

Lehmer's question is equivalent to one about generalized growth rates of Lefschetz numbers of iterated pseudo-Anosov surface homeomorphisms. One need consider only homeomorphisms that arise as monodromies of fibered knots in lens spaces L(n,1), n>0. Lehmer's question for Perron polynomials is equivalent to one about ge…

Study on endomorphism and automorphism groups of specific quandles.

Any endomorphism of a finitely generated free group naturally descends to an injective endomorphism of its stable quotient. In this paper, we prove a geometric incarnation of this phenomenon: namely, that every expanding irreducible train track map inducing an endomorphism of the fundamental group gives rise to an expa…

We determine the automorphisms and the continuous endomorphisms of the Einstein gyrogroup in arbitrary dimension. This generalizes a recent result of Lajos Molnár and Dániel Virosztek, who have determined the continuous endomorphisms in the three-dimensional case.

In the description of the instanton Floer homology of a surface times a circle due to Muñoz, we compute the nilpotency degree of the endomorphism $u^2-64$. We then compute the framed instanton homology of a surface times a circle with non-trivial bundle, which is closely related to the kernel of $u^2-64$. We discuss th…

We prove that the mass endomorphism associated to the Dirac operator on a Riemannian manifold is non-zero for generic Riemannian metrics. The proof involves a study of the mass endomorphism under surgery, its behavior near metrics with harmonic spinors, and analytic perturbation arguments.

The monoids of simplicial endomorphisms, i.e. the monoids of endomorphisms in the simplicial category, are submonoids of monoids one finds in Temperley-Lieb algebras, and as the monoids of Temperley-Lieb algebras are linked to situations where an endofunctor is adjoint to itself, so the monoids of simplicial endomorphi…

By applying a variant of the TQFT constructed by Blanchet, Habegger, Masbaum, and Vogel, and using a construction of Ohtsuki, we define a module endomorphism for each knot K by using a tangle obtained from a surgery presentation of K. We show that it is strong shift equivalent to the Turaev-Viro endomorphism associated…

We give a necessary and sufficient condition on the 1-jet of a field of nilpotent endomorphisms to be integrable. Together with the well known corresponding condition for an almost complex structure, the nullity of its Nijenhuis tensor, this gives an integrability condition for any field of endomorphisms.

Defines strongest integrability condition for skew-symmetric endomorphisms.

Study character varieties of tangles to map immersed curves in the pillowcase.

New quadratic forms expand and rotate linear endomorphisms in geometric theory.

We construct expanding endomorphisms on smooth manifolds that are homeomorphic to tori yet have exotic underlying PL-structures.

Paper establishes an isomorphism between Fukaya category and bordered knot Floer homology.

This paper proves all endomorphisms of framed little disk operad are automorphisms.

Novel ternary structures reveal new interpretations of linear connections.

We prove that the outer automorphism group $Out(G)$ is residually finite when the group $G$ is virtually compact special (in the sense of Haglund and Wise) or when $G$ is isomorphic to the fundamental group of some compact $3$-manifold. To prove these results we characterize commensurating endomorphisms of acylindrical…

New polynomial invariants for knots and links.

We study Nijenhuis structures on Courant algebroids in terms of the canonical Poisson bracket on their symplectic realizations. We prove that the Nijenhuis torsion of a skew-symmetric endomorphism N of a Courant algebroid is skew-symmetric if the square of N is proportional to the identity, and only in this case when t…

Let φbe an endomorphism of a finitely generated free group F, and let H be a finite-index subgroup of F that is invariant under φ. The nonzero eigenvalues of φare contained in the eigenvalues of φrestricted to H.

Detects free group automorphisms using homology of covers.

Study of transitivity in partially hyperbolic maps with expanding linear part.

Hyperbolicity proven for a specific type of group extension.

A fixed point theorem is proved for inverse transducers, leading to an automata-theoretic proof of the fixed point subgroup of an endomorphism of a finitely generated virtually free group being finitely generated. If the endomorphism is uniformly continuous for the hyperbolic metric, it is proved that the set of regula…

Let $(M,g)$ be a closed Riemannian spin manifold. The constant term in the expansion of the Green function for the Dirac operator at a fixed point $p\in M$ is called the mass endomorphism in $p$ associated to the metric $g$ due to an analogy to the mass in the Yamabe problem. We show that the mass endomorphism of a gen…

The Complex Axis theorem states that any endomorphism of a finite-dimensional complex vector space affords an eigen-vector (or "invariant axis"). A geometric proof of this geometric result was given by A. de Medeiros, transforming the endomorphism into a topological self-map with Lefschetz number not equal to zero. We …

We construct an endomorphism of the Khovanov invariant to prove H-thinness and pairing phenomena of the invariants for alternating links. As a consequence, it follows that the Khovanov invariant of an oriented nonsplit alternating link is determined by its Jones polynomial, signature, and the linking numbers of its com…

Study big mapping class groups and their co-Hopfian property, finding new examples and proving injective homomorphisms results.

Classifies 2-solvable Frobenius Lie algebras based on endomorphisms.

A new method classifies almost contact metric manifolds using intrinsic endomorphisms.

In this letter we investigate some aspects of the noncommutative differential geometry based on derivations of the algebra of endomorphisms of an oriented complex hermitian vector bundle. We relate it, in a natural way, to the geometry of the underlying principal bundle and compute the cohomology of its complex of nonc…

The Magnus representation of the Torelli subgroup of the mapping class group of a surface is a homomorphism r: I_{g,1} -> GL_{2g}(Z[H]). Here H is the first homology group of the surface. This representation is not faithful; in particular, Suzuki previously described precisely when the commutator of two Dehn twists abo…

We study noncommutative generalizations of such notions of the classical symplectic geometry as degenerate Poisson structure, Poisson submanifold and quotient manifold, symplectic foliation and symplectic leaf for associative Poisson algebras. We consider these structures for the case of the endomorphism algebra of a v…