A group is said to have the R∞ property if every automorphism has an infinite number of twisted conjugacy classes. We study question whether G when a finitely generated torsion-free nilpotent group. As consequence, we show that for positive integer n ≧ 5, there compact nilmanifold dimension on which homeomorphism isotopic fixed point free homeomorphism. by-product, give purely theoretic proof two generators property. The virtually abelian and -nilpotent groups are also discussed.

Let $\Sigma_{g,p}$ be an orientable surface of genus $g$ and finite type without boundary (i.e. closed with a number $p$ points removed). In this paper we study the R$_{\infty}$-property for pure braid groups $P_n(\Sigma_{g,p})$ as well full $B_n(\Sigma_{g,p})$. We show that, few exceptions, these have R$_{\infty}$-property.

Let φ:G → G be a group endomorphism where is finitely generated of exponential growth, and denote by R(φ) the number twisted φ-conjugacy classes. Fel'shtyn Hill (K-theory 8 (1994) 367–393) conjectured that if φ injective, then infinite. This paper shows this conjecture does not hold in general. In fact, can finite for some automorphism φ. Furthermore, certain family polycyclic groups, there no injective with R(φn) < ∞ all n. 2000 Mathematics Subject Classification 20E45, 37C25, 20F16, 55M20, 20F99.

Let G be a finitely generated abelian group and ≀ ℤ the wreath product. In this paper, we classify all such groups for which every automorphism of has infinitely many twisted conjugacy classes.

Let . This generalises results concerning the sphere of Gillette, Van Buskirk and Murasugi. We also show that Artin pure braid groups admit a (non-trivial) splitting as direct product which one factors is cyclic group generated by full twist.

Using Sigma theory we show that for large classes of groups G there is a subgroup H finite index in Aut(G) such ϕ ∈ the Reidemeister number R(ϕ) infinite.This includes all finitely generated nonpolycyclic fall into one following classes: nilpotent-by-abelian type FP ∞ ; G/ Prüfer rank; 2 without free nonabelian subgroups and with maximal metabelian quotient; some direct products groups; or pure symmetric automorphism group.Using different argument result also holds 1-ended nonsurface limit...

Let B n (RP 2 ) (respectively P )) denote the braid group pure group) on strings of real projective plane RP .In this paper we study these groups, in particular associated short exact sequence Fadell and Neuwirth, their torsion elements roots 'full twist' braid.Our main results may be summarised as follows: first, 1, there is a k -torsion element if only divides either 4n or 4(n -1).Finally, full twist has th root 2n 2(n -1).

Our aim is to determine the lower central series (LCS) and derived (DS) for braid groups of sphere finitely-punctured sphere. We show that all n (resp. n\geq 5), LCS DS) n-string group B\_n(S^2) constant from commutator subgroup onwards, \Gamma\_2(B\_4(S^2)) a semi-direct product quaternion by free rank 2. For n=4, we DS B\_4(S^2), as well its quotients. \geq 1, class m-string B\_m(S^2) \ {x\_1,...,x\_n} n-punctured includes Artin B\_m, those annulus, certain affine groups. extend results...

Let F be either a free nilpotent group of given class and finite rank or solvable certain derived length rank. We show precisely which ones have the R ∞ property. Finally, we also that infinite does not

In this article, we prove that any automorphism of R. Thompson's group F has infinitely many twisted conjugacy classes.The result follows from the work Brin, together with standard facts about F, and elementary properties Reidemeister numbers.

Let φ : → be an automorphism of a group .We say that x, y ∈ are in the same φ-twisted conjugacy class and write x ∼ if there exists element γ such = xφ(γ -1 ).This is equivalence relation on called conjugacy.Let R(φ) denote number classes .If infinite for all Aut( ), we has R ∞ -property.The purpose this note to show symmetric S , Houghton groups pure have -property.We show, also, Richard Thompson T obtain general result establishing -property finite direct product finitely generated...

For maps f, g : X → Y between closed orientable n-manifolds, we investigate conditions for which N(f, g) = R(f, where and denote the Nielsen Reidemeister coincidence numbers respectively. In particular, give necessary suffcient equality to hold when is a solvmanifold or both are infrasolvmanifolds.

We classify the (finite and infinite) virtually cyclic subgroups of pure braid groups Pn(ℝP2) projective plane. The maximal finite are isomorphic to quaternion group order 8 if n = 3, ℤ4 ⩾ 4. Further, for all following are, up isomorphism, infinite Pn(ℝP2): ℤ, ℤ2 × ℤ amalgamated product ∗ℤ2 ℤ4.

We generalize the Lefschetz coincidence theorem to non-oriented manifolds. use (co-) homology groups with local coefficients. This generalization requires assumption that one of considered maps is orientation true.

Given a pair (S, T) whereS is closed surface and T free Z 2 action on S, we classify which pairs have the property that Borsuk-Ulam theorem holds with respect to R 2. In particular if S orientable its Euler characteristic congruent mod 4, this always case independent of action. We say for any map f : → there point x ∊ such f(x) = f(T(x)).