Abstract. Given a group automorphism φ : Γ −→ Γ, one has an action of Γ on itself by φ-twisted conjugacy, namely, g.x = gxφ(g −1 ). The orbits of this action are called φ-twisted conjugacy classes. One says that Γ has the R ∞ -property if there are infinitely many φ-twisted conjugacy classes for every automorphism φ of Γ. In this paper we show that SL(n, Z) and its congruence subgroups have the R ∞ -property. Further we show that any (countable) abelian extension of Γ has the R ∞ -property where Γ is a torsion free non-elementary hyperbolic group, or SL(n, Z), Sp(2n, Z) or a principal congruence subgroup of SL(n, Z) or the fundamental group of a complete Riemannian manifold of constant negative curvature.

Abstract. Given a group automorphism φ : Γ −→ Γ, one has an action of Γ on itself by φ-twisted conjugacy, namely, g.x = gxφ(g −1 ). The orbits of this action are called φ-conjugacy classes. One says that Γ has the R ∞ -property if there are infinitely many φ-conjugacy classes for every automorphism φ of Γ. In this paper we show that any irreducible lattice in a connected non-compact semi simple Lie group having finite centre and rank at least 2 has the R ∞ -property.

Let G be a non-compact semisimple Lie group with finite centre and finitely many connected components. We show that any finitely generated group Γ which is quasi-isometric to an irreducible lattice G has the R ∞ -property, namely, that there are infinitely many φ-twisted conjugacy classes for every automorphism φ of Γ. Also, we show that any lattice in G has the R ∞ -property, extending our earlier result for irreducible lattices.

In this article, we prove that the group of all increasing homeomorphisms on R has exactly five normal subgroups, and the group of all homeomorphisms on R has exactly four normal subgroups. There are several results known about the group of homeomorphisms on R and about the group of increasing homeomorphisms on R ([2], [6], [7] and [8]), but beyond this there is virtually nothing in the literature concerning the topological structure in the aspects of topological dynamics. In this paper, we analyze this structure in some detail.

Several interesting numbers such as the homotopy invariant numbers the Lefschets number L(f), the Nielsen number N(f), fixed point index i(X, f,U) and the Reidemeister number R(f) play important roles in the study of fixed point theorems. The Nielsen number gives more geometric information about fixed points than other numbers. However the Nielsen number is hard to compute in general. To compute the Nielsen number, Jiang related it to the Reidemeister number R(f ) of the induced homomorphism f  :  1(X)    1(X) when X is a lens space or an H-space (Jian type space). For such spaces, either N(f) = 0 or N(f) = R(f) the Reidemeister number of f  and if R(f) =   then N(f) = 0 which implies that f is homotopic to a fixed point free map. This is a review article to discuss how these numbers are related in fixed point theory.