Related papers: Finite groups acting symplectically on $T^2\times …

A recent preprint of Csik\'os, Pyber and Szab\'o (arXiv:1411.7524) proves that the diffeomorphism group of $T^2\times S^2$ is not Jordan. The purpose of this paper is to generalize the arguments of Csik\'os, Pyber and Szab\'o in order to…

Let H_g denote the closed 3-manifold obtained as the connected sum of g copies of S^2 times S^1, with free fundamental group of rank g. We prove that, for a finite group G acting on H_g which induces a faithful action on the fundamental…

The purpose of this article is to characterize symplectic and Hamiltonian circle actions on symplectic manifolds in terms of symplectic embeddings of Riemann surfaces. More precisely, we will show that (1) if $(M,\omega)$ admits a…

Let $M=(M,\omega)$ be either the product $S^2\times S^2$ or the non-trivial $S^2$ bundle over $S^2$ endowed with any symplectic form $\omega$. Suppose a finite cyclic group $Z_n$ is acting effectively on $(M,\omega)$ through Hamiltonian…

We prove that for any closed smooth $4$-manifold $X$ there exists a constant $C$ with the property that each finite subgroup $G<Diff(X)$ has a subgroup $N$ which is abelian or nilpotent of class $2$, and which satisfies $[G:N]\leq C$. We…

We study generators of the fundamental group of the group of symplectomorphisms $\mathrm{Symp}({\mathbb C\mathbb P}^2\#\,5\overline{\mathbb C\mathbb P}\,\!^2, \omega)$ for some particular symplectic forms. It was observed by J. K\c{e}dra…

A group is SimpHAtic if it acts geometrically on a simply connected simplicially hereditarily aspherical (SimpHAtic) complex. We show that finitely presented normal subgroups of the SimpHAtic groups are either: finite, or of finite index,…

We consider the 3-point blow-up of the manifold $ (S^2 \times S^2, \sigma \oplus \sigma)$ where $\sigma$ is the standard symplectic form which gives area 1 to the sphere $S^2$, and study its group of symplectomorphisms $\rm{Symp} ( S^2…

From the cohomological point of view the symplectomorphism group $Sympl (M)$ of a symplectic manifold is `` tamer'' than the diffeomorphism group. The existence of invariant polynomials in the Lie algebra $\frak {sympl }(M)$, the symplectic…

If $G_1$ and $G_2$ are finite groups with periodic Tate cohomology, then $G_1\times G_2$ acts freely and smoothly on some product $S^n \times S^n$.

We construct a non-abelian extension $\Gamma$ of $S^1$ by $\cy 3 \times \cy 3$, and prove that $\Gamma$ acts freely and smoothly on $S^{5} \times S^{5}$. This gives new actions on $S^{5} \times S^{5}$ for an infinite family $\cP$ of finite…

We prove new cases of the Hilbert-Smith conjecture for actions by natural homeomorphisms in symplectic topology. Specifically, we prove that the group of $p$-adic integers $\mathbb Z_p$ does not admit non-trivial continuous actions by…

We construct smooth symplectic resolutions of the quotient of R^2 under some infinite discrete sub-group of GL_2(R) preserving a log-symplectic structure. This extends from algebraic geometry to smooth real differential geometry the Du Val…

Let $p$ be an odd prime. We construct a non-abelian extension $\Gamma$ of $S^1$ by $Z/p \times Z/p$, and prove that any finite subgroup of $\Gamma$ acts freely and smoothly on $S^{2p-1} \times S^{2p-1}$. In particular, for each odd prime…

We prove that, for nice classes of infinite-dimensional smooth groups G, natural constructions in smooth topology and symplectic topology yield homotopically coherent group actions of G. This yields a bridge between infinite-dimensional…

Let $(S^2,\omega)$ be a symplectic sphere, and let $\tau \colon S^2 \to S^2$ be an anti-symplectic involution of $(S^2,\omega)$. We consider the product $(S^2,\omega) \times (S^2,\omega)$ endowed with the anti-symplectic involution $\tau…

Let $X$ be the $2$-sphere $\mathbb S^2$ or the real projective plane $\mathbb {RP}^2$. We show that if $\Gamma$ is a finitely generated group acting minimally and distally on $X$, then $\Gamma$ contains a nonabelian free subgroup.

Let (M,\omega) be a four dimensional compact connected symplectic manifold. We prove that (M,\omega) admits only finitely many inequivalent Hamiltonian effective 2-torus actions. Consequently, if M is simply connected, the number of…

We study the extension of homologically trivial symplectic or Hamiltonian cyclic actions to Hamiltonian circle actions on irrational ruled symplectic $4$-manifolds. On one hand, we construct symplectic involutions on minimal irrational…

For groups of diffeomorphisms of $\T^2$ containing an Anosov diffeomorphism, we give a complete classification for polycyclic Abelian-by-Cyclic group actions on $\T^2$ up to both topological conjugacy and smooth conjugacy under mild…