In this paper, we show that every element of a discrete module is a sum of two units if and only if its endomorphism ring has no factor ring isomorphic to $Z_{2}$. We also characterize unit sum number equal to two for the endomorphism ring of quasi-discrete modules with finite exchange property.

In this paper, we consider the Frobenius endomorphism on twisted Edwards curve and give the characteristic polynomial of the map. Applying the Frobenius endomorphism on twisted Edwards curve, we construct a skew-Frobenius map defined on the quadratic twist of an twisted Edwards curve. Our results show that the Frobenius endomorphism on twisted Edwards curve and the skew-Frobenius endomorphism o...

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 acylindrically hyperbolic groups. An endomorphism φ of a group G is said to be commensurating, if...

Recall that if (K, v) is a valued field, one may consider its completion (K̂, v̂). The extension K̂/K is immediate, i.e. kK = kK̂ and ΓK = ΓK̂ . Definition. • A valued difference field is a structure (K, v, σ), where (K, v) is a valued field and σ is a field endomorphism satisfying σ(x) ∈ OK iff x ∈ OK . It induces a field endomorphism σk on kK and an endomorphism (of ordered abelian groups) σΓ on ΓK .

We prove that a C orientation-preserving circle endomorphism which is Hölder conjugate to a C circle expanding endomorphism is itself expanding.

Infinite words over a finite special confluent rewriting system R are considered and endowed with natural algebraic and topological structures. Their geometric significance is explored in the context of Gromov hyperbolic spaces. Given an endomorphism φ of the monoid generated by R, existence and uniqueness of several types of extensions of φ to infinite words (endomorphism extensions, weak endo...

Let $f \colon X \to X$ be a surjective endomorphism of normal projective surface. When $\operatorname{deg} f \geq 2$, applying an (iteration of) $f$-equivariant minimal model program (EMMP), we determine the geometric structure $X$. Using this, extend second author's result to singular surfaces extent that either $X$ has $f$-invariant non-constant rational function, or $f$ infinitely many Zaris...

This paper considers the following question: Given an Anosov endomorphism f on T, whether f is topologically conjugate to some hyperbolic toral endomorphism? It is well known that the answer for Anosov diffeomorphisms and expanding endomorphisms is affirmative. However, for the remainder Anosov endomorphisms, a quite different answer is obtained in this paper, i.e., for generic Anosov endomorph...

in this paper, we show that every element of a discrete module is a sum of two units if and only if its endomorphism ring has no factor ring isomorphic to $z_{2}$. we also characterize unit sum number equal to two for the endomorphism ring of quasi-discrete modules with finite exchange property.