Related papers: The domino problem is undecidable on surface group…

In this paper we prove that the Diophantine problem in iterated restricted wreath products $G$ of arbitrary non-trivial free abelian groups $A_1,\ldots, A_k$, $k>1$ of finite ranks is undecidable, i.e., there is no algorithm that given a…

We prove that on closed Riemannian manifolds with infinite abelian, but not cyclic, fundamental group, any isometry that is homotopic to the identity possesses infinitely many invariant geodesics. We conjecture that the result remains true…

The description of invariants of surfaces with respect to the motion groups is reduced to the description of invariants of parameterized surfaces with respect to the motion groups. Existence of a commuting system of invariant partial…

The paper contains a general construction which produces new examples of non simply-connected smooth projective surfaces. We analyze the resulting surfaces and their fundamental groups. Many of these fundamental groups are expected to be…

We solve a certain case of the minimal genus problem for embedded surfaces in elliptic 4-manifolds. The proofs involve a restricted transitivity property of the action of the orientation preserving diffeomorphism group on the second…

We prove an algebraic version of a classical theorem in topology, asserting that an abelian p-group action on a smooth projective variety of positive dimension cannot fix exactly one point. When the group has only two elements, we prove…

We prove that various classical conformal diffeomorphism groups, which are known to be essential [1], are in fact properly essential. This is a consequence of a local criterion on a conformal diffeomorphism in the form of a cohomological…

This paper completely determines the non-amenability of the mapping class groups of infinite-type surfaces, the mapping class groups of locally finite infinite graphs of higher ranks, gives an example of non-amenable stabiliser of a point…

We consider a helicoidal group $G$ in $\mathbb{R}^{n+1}$ and unbounded $G$-invariant $C^{2,\alpha}$-domains $\Omega\subset\mathbb{R}^{n+1}$ whose helicoidal projections are exterior domains in $\mathbb{R}^{n}$, $n\geq2$. We show that for…

Questions related to deformations of germs of finite morphisms of smooth surfaces are discussed. A classification of the four-sheeted germs of finite covers $F: (U,o')\to (V,o)$ is given up to smooth deformations, where $(U,o')$ and $(V,o)$…

We prove a result that relates the number of homomorphisms from the fundamental group of a compact nonorientable surface to a finite group $G$, where conjugacy classes of the boundary components of the surface must map to prescribed…

One can embed arbitrarily many disjoint, non-parallel, non-boundary parallel, incompressible surfaces in any three manifold with at least one boundary component of genus two or greater [4]. This paper proves the contrasting, but not…

The existence problem for holomorphic structures on vector bundles over non-algebraic surfaces is in general still open. We solve this problem in the case of rank 2 vector bundles over K3 surfaces and in the case of vector bundles of…

Tiling planar regions with dominoes is a classical problem in which the decision and counting problems are polynomial. We prove a variety of hardness results (both NP- and #P-completeness) for different generalizations of dominoes in three…

We study slope stability of smooth surfaces and its connection with exceptional divisors. We show that a surface containing an exceptional divisor with arithmetic genus at least two is slope unstable for some polarisation. In the converse…

Using a recent result of Bowden, Hensel and Webb, we prove the existence of homeomorphisms with positive stable commutator length in the groups of homeomorphisms of the real projective plane and M\"obius strip which are isotopic to the…

We show that the identity component of the group of diffeomorphisms of a closed oriented surface of positive genus admits many unbounded quasi-morphisms. As a corollary, we also deduce that this group is not uniformly perfect and its…

We investigate the problem of deciding whether a system of linear equations, together with divisibility conditions on the variables, has a solution over holomorphy subrings of global fields. We obtain decidability results when we allow…

Aperiodic tilings are non-periodic tilings characterized by local constraints. They play a key role in the proof of the undecidability of the domino problem (1964) and naturally model quasicrystals (discovered in 1982). A central question…

In this short expository note, we give a detailed proof of Markov's theorem on the unsolvability of the homeomorphism problem and of the existence of unrecognizable manifolds in all dimensions larger than 3.