In this short note, we study the bordism problem for group actions on torus and give examples of groups acting by diffeomorphisms isotopic to identity that cannot be extended an action a bounding 3-manifold. This solves question raised in previous work authors.

This can be stated in a more symmetric manner. Let r be any positive integer not equal to 3. Then n acts freely and homologically trivially on Z r i ff n acts freely and homologically trivially on SL In fact, there is a one-to-one correspondence between such actions on U and such actions on S r. (The classification of such actions is discussed in w In addition the actions constructed have the p...

We consider the entropy of systems of random transformations, where the transformations are chosen from the set of generators of a Z action. We show that the classical definition gives unsatisfactory entropy results in the higher-dimensional case, i.e. when d ≥ 2. We propose a new definition of the entropy for random group actions which agrees with the classical definition in the one-dimensiona...

We address the following question: Determine the affine cones over smooth projective varieties which admit an action of a connected algebraic group different from the standard C∗-action by scalar matrices and its inverse action. We show in particular that the affine cones over anticanonically embedded smooth del Pezzo surfaces of degree ≥ 4 possess such an action. A question in [FZ1] whether th...

We investigate the geometry of median metric spaces. The group-theoretic applications are towards Kazhdan’s property (T) and Haagerup’s property.

We set S1 = R/2πZ throughout. Let (M,ω) be a symplectic manifold. A symplectic S1-action on (M,ω) is a smooth family ψt ∈ Symp(M,ω), t ∈ S1, such that ψt+s = ψt ◦ ψs for any t, s ∈ S1. One can easily check that the corresponding vector fields Xt ≡ d dtψt ◦ψ −1 t is time-independent, i.e., Xt = X is constant in t. We call X the associated vector field of the given symplectic S1-action. Note that...

Given a finite group G acting as automorphisms on a ring A, the skew group ring A∗G is an important tool for studying the structure of G-stable ideals of A. The ring A∗G is G-graded, i.e.G coacts on A∗G. The Cohen-Montgomery duality says that the smash product A ∗ G#k[G]∗ of A ∗ G with the dual group ring k[G]∗ is isomorphic to the full matrix ring Mn(A) over A, where n is the order of G. In th...

We develop the theory of rational ideals for arbitrary associative algebras R without assuming the standard finiteness conditions, noetherianness or the Goldie property. The Amitsur-Martindale ring of quotients replaces the classical ring of quotients which underlies the previous definition of rational ideals but is not available in a general setting. Our main result concerns rational actions o...