arXiv research

Novel ternary structures reveal new interpretations of linear connections.

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

Absolutely partially hyperbolic surface endomorphisms have a coherent center foliation.

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

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 …

Constructs Lepage equivalents for arbitrary-order Lagrangians.

Computes $L_\infty$-algebroid for linear foliations on vector spaces.

In this paper, we give complete classifications of linear $\infty$-harmonic maps between Euclidean and Heisenberg spaces, between Nil and Sol spaces. We also classify all $\infty$-harmonic linear endomorphisms of Sol space and show that there is a subgroup of $\infty$-harmonic linear automorphisms in the group of linea…

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…

Let $M$ be a closed manifold that admits a self-cover $p:M \to M$ of degree >1. We say p is strongly regular if all its iterates are regular covers. In this case, we establish an algebraic structure theorem for the fundamental group of $M$: We prove that $π_1(M)$ surjects onto a nontrivial free abelian group $A$, and t…

Quotients of torus endomorphisms have parabolic orbifolds.

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

Study on endomorphism and automorphism groups of specific quandles.

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

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.

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.

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

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

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…

Generalization of the cross ratio to polarizations of linear finite and infinite-dimensional spaces (in particular to Sato Grassmannian) is given and explored. This cross ratio appears to be a cocycle of the canonical (tautalogical) bundle over the Grassmannian with coefficients in the sheaf of its endomorphisms. Opera…

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 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…

Proves torsion and curvature are unique for smooth manifolds.

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.

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…

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 …

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

Detects free group automorphisms using homology of covers.

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 fixed point indices and words at infinity for graph selfmaps.

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…

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…

We study the fields of endomorphisms intertwining pairs of symplectic structures. Using these endomorphisms we prove an analogue of Moser's theorem for simultaneous isotopies of two families of symplectic forms. We also consider the geometric structures defined by pairs and triples of symplectic forms for which the squ…

Let M be a compact manifold equipped with a Riemannian metric g and a spin structure \si. We let $λ(M,[g],\si)= \inf_{\tilde{g} \in [g]} λ_1^+(\tilde{g}) Vol(M,\tilde{g})^{1/n}$ where $λ_1^+(\tilde{g})$ is the smallest positive eigenvalue of the Dirac operator D in the metric $\tilde{g}$. A previous result stated that …

Study describes signatures of Ricci curvature on nilmanifolds.

We prove that for certain endomorphisms of a nilmanifold N the set S of those points such that the closure of its (forward) orbit contains no periodic points is large in the sense that for any non-empty open set U, the set U\cap S is of full Hausdorff dimension. When the manifold N is a torus, this result is due to S.G…

The paper derives a formula for Lefschetz number of a geometric endomorphism.