FUNDAMENTAL GROUPS OF CLOSED POSITIVELY CURVED MANIFOLDS WITH ALMOST DISCRETE ABELIAN GROUP ACTIONS

Let M be a closed n-manifold of positive sectional curvature. Assume that M admits an effective isometrical T1× Zpk-action with p prime. The main result of the article n＋1 for n 〉 5, then there exists a positive constant p（n）, is that ifk=lforn=3or k〉 n＋1/4 for n≥5,then there exists a positive constant p（n）,depending only on n, such that π1 （M） is cyclic if p ≥ p（n）.

Bounding Topology via Geometry,A-Simple Fundamental Groups

We call a group A-simple,if it has no non-trivial normal abelian subgroup.We will present finiteness results in controlled topology via geometry on manifolds whose fundamental groups are A-simple.

Fundamental Groups of Graphs of Cyclic Subgroup Separable and Weakly Potent Groups

We give a characterization of the cyclic subgroup separability and weak potency of the fundamental group of a graph of polycyclic-by-finite groups and free-by-finite groups amalgamating edge subgroups of the form<h>×D,where h has infinite order and D is finite.

Fundamental Groups of Triangle Geometries

In this paper , depending on the theory of buildings, we obtain a kind of method completely different from past ones, to compute topological fundamental groups of some triangle geometries. This method will enable us to easily compulte topological fundametal groups of infinitely finite triangle geometries.

On fundamental groups of Finsler manifolds In memory of the centenary birthday of Professor S.S.Chern

In this paper,we study the growth of fundamental groups of Finsler manifolds.Some relationships between the growth of fundamental groups and the volume growth of universal covers of Finsler manifolds are found.Some estimates of entropies and the number of generators of fundamental groups of Finsler manifolds are given.Moreover,the quasi-isometry and the geometric norm in Finsler geometry are considered.

Fundamental Groups of Real Toric Spaces over Simple Polytopes

In this paper we study the cubical stuctures and fundamental groups of real toric spaces.We give an explicit presentation of the fundamental group of the real toric space over a simple polytope.Then using this presentation,we give a description of the existence of non-degenerate colourings on a simple polytope from a homotopy point of view.

The topological entropy for autonomous Lagrangian systems on compact manifolds whose fundamental groups have exponential growth

In this article,we consider the topological entropy for autonomous positive definite Lagrangian systems on connected closed Riemannian manifolds whose fundamental groups have exponential growth.We prove that on each energy level E(x,v)=k with k>c(L),where c(L)is Mane’s critical value,the EulerLagrange flow has positive topological entropy.This extends the classical Dinaburg theorem from geodesic flows to general autonomous positive definite Lagrangian systems.

HARMONIC MAPS AND FUNDAMENTAL GROUPS OF NONPOSITIVELY CURVEDRIEMANNIAN MANIFOLDS

Using the theory of harmonic maps the authors discuss theproperties of the fundamental group of a complete nonpositivelycurved Riemannian manifold, and prove that the finitely generatedvirtual solvable subgroup of fundamental group of a completenonpositively curved Riemannian manifold either is a peripheralsubgroup of fundamental group or can be realized by animmersed totall geodesic closed flat manifold. It generalizessome results of Gromoll-Wolf, Lawson-Yan and Schoen-Yau.

Generalized Baumslag-Solitar Graphs with Soluble Fundamental Groups

It is shown that the generalized Baumslag-Solitar graphs which have soluble fundamental groups can be constructed from four simple types of graph by a process of decoration. The method of proof is elementary and employs the operations of pinching and contracting edges of graphs.

On Fundamental Group of a Certain Class of Welded Knots

In this paper, a certain class of welded knots K;is considered. By calculating the commutators subgroup of fundamental group Gn of welded knot K;,n ∈ Z;, we show that these welded knots are not equivalent to each other and they are all not classical knots. Secondly, we study some properties of Gn and obtain that Gn is linear, residually finite and Hopfian.

Orbifold Fundamental Group and Deck Translation Group

In this paper, we define the notions of orbifold loop and orbifold road, with which, we reformulate the definition of orbifold fundamental group and deck translation group. We show that, for each orbifold, the orbifold fundamental group is isomorphic to the deck translation group.

The Homotopical Proof of Π₁(<i>S</i>, <i>x_o</i>) as a Fundamental Group in a General Interval

The aim of this study is to establish that, the equivalent class which is made up of homotopic loops is a group with respect to in the general interval [m,n]. The study proved from homotopical point of view that is associative, has an identity and inverse function. The study established with proof that is a fundamental group in [m,n] ,.

Lower bounds of the topological entropy for nonautonomous dynamical systems

In this paper, lower bounds of the topological entropy for nonautonomous dynamical systems are given via the growths of topological complexity in fundamental group and in degree.

Minimal Euler characteristics of 4-manifolds with 3-manifold groups

Letπ=π_(1)(M)for a compact 3-manifold M,andχ_(4),p and q^(*)be the invariants of Hausmann and Weinberger(1985),Kotschick(1994)and Hillman(2002),respectively.For a certain class of compact 3-manifolds M,including all those not containing two-sided RP2’s,we determineχ_(4)(π).We address when p(π)equalsχ_(4)(π)and when q^(*)(π)equalsχ_(4)(π),and answer a question raised by Hillman(2002).

ON THE FUNDAMENTAL GROUP OF OPEN MANIFOLDS WITH NONNEGATIVE RICCI CURVATURE

The authors establish some uniform estimates for the distance to halfway points of minimalgeodesics in terms of the distantce to end points on some types of Riemannian manifolds, andthen prove some theorems about the finite generation of fundamental group of Riemannianmanifold with nonnegative Ricci curvature, which support the famous Milnor conjecture.

Ricci Curvature and Fundamental Group

Abstract By refined volume estimates in terms of Ricci curvature, the two results due to J. Milnor （1968） are generalized.

The Fundamental Group of the Complement of the Branch Curve of CP^1×T

Denoting by T the complex projective torus, we can embed the surface CP^1 × T in CP^5. In this paper we compute the fundamental group of the complement of the branch curve of this surface. Since the embedding is not ＂ample enough＂, the embedded surface does not belong to the classes of surfaces where the fundamental group is virtually solvable： a property which holds for these groups for ＂ample enough＂ embeddings. On the other hand, as it is the first example of this computation for non simply-connected surfaces, the structure of this group （as shown in this paper） give rise to the extension of the conjecture regarding the structure of those fundamental groups of any surface.

ON THE SHEAF OF GENERALIZED FUNDAMENTAL GROUP AND COVERING SPACE

The author constructs the sheaf of generalized fundamental group of a topological space Xand obtains the relationship between its lst cohomology set and all covering spaces Of X undersuitable conditions imposed upon X.

Compact n-Manifolds via(n+l)-Colored Graphs:a New Approach

We introduce a representation via(n+l)-colored graphs of compact n-manifolds with(possibly empty)boundary,which appears to be very convenient for computer aided study and tabulation.Our construction is a generalization to arbitrary dimension of the one recently given by Cristofori and Mulazzani in dimension three,and it is dual to the one given by Pezzana in the 1970s.In this context we establish some results concerning the topology of the represented manifolds:suspensions,fundamental groups,connected sums and moves between graphs representing the same manifold.Classification results of compact orientable 4-manifolds representable by graphs up to six vertices are obtained,together with some properties of the G-degree of 5-colored graphs relating this approach to tensor models theory.

A note on Higgs-de Rham flows of level zero

The notion of Higgs-de Rham flows was introduced by Lan et al.(2019),as an analogue of Yang-Mills-Higgs flows in the complex nonabelian Hodge theory.In this paper we investigate a small part of this theory,and study those Higgs-de Rham flows which are of level zero.We improve the original definition of level-zero Higgs-de Rham flows(which works for general levels),and establish a Hitchin-Simpson type correspondence between such objects and certain representations of fundamental groups in positive characteristic,which generalizes a classical results of Katz(1973).We compare the deformation theories of two sides in the correspondence,and translate the Galois action on the geometric fundamental groups of algebraic varieties defined over finite fields into the Higgs side.