Piotr Przytycki

Separability of embedded surfaces in 3-manifolds

We prove that if S is a properly embedded incompressible surface in a compact 3-manifold M, then the fundamental group of S is separable in the fundamental group of M.

Realisation and dismantlability

We prove that a finite group acting on an infinite graph with dismantling properties fixes a clique. We prove that in the flag complex spanned on such a graph the fixed point set is contractible. We study dismantling properties of the arc, disc and sphere graphs. We apply our theory to prove that any finite subgroup H of the mapping class group of a surface with punctures, the handlebody group, or Out.Fn/ fixes a filling (respectively simple) clique in the appropriate graph. We deduce some realisation theorems, in particular the Nielsen realisation problem in the case of a nonempty set of punctures. We also prove that infinite H have either empty or contractible fixed point sets in the corresponding complexes. Furthermore, we show that their spines are classifying spaces for proper actions for mapping class groups and Out.Fn/. 20F65

Contractibility of the Kakimizu complex and symmetric Seifert surfaces

Kakimizu complex of a knot is a flag simplicial complex whose vertices correspond to minimal genus Seifert surfaces and edges to disjoint pairs of such surfaces. We discuss a general setting in which one can define a similar complex. We prove that this complex is contractible, which was conjectured by Kakimizu. More generally, the fixed-point set (in the Kakimizu complex) for any subgroup of an appropriate mapping class group is contractible or empty. Moreover, we prove that this fixed-point set is non-empty for finite subgroups, which implies the existence of symmetric Seifert surfaces.

Acute triangulations of polyhedra and ℝ N

We study the problem of acute triangulations of convex polyhedra and the space ℝn. Here an acute triangulation is a triangulation into simplices whose dihedral angles are acute. We prove that acute triangulations of the n-cube do not exist for n≥4. Further, we prove that acute triangulations of the space ℝn do not exist for n≥5. In the opposite direction, in ℝ3, we present a construction of an acute triangulation of the cube, the regular octahedron and a non-trivial acute triangulation of the regular tetrahedron. We also prove nonexistence of an acute triangulation of ℝ4 if all dihedral angles are bounded away from π/2.

Cocompactly cubulated 2-dimensional Artin groups

We give a necessary and sufficient condition for a 2-dimensional or a three-generator Artin group

Twist-rigid Coxeter groups

A

The ending lamination space of the five-punctured sphere is the Nöbeling curve

to be (virtually) cocompactly cubulated, in terms of the defining graph of

Bipolar Coxeter groups

A

EG for systolic groups

.

The fixed point theorem for simplicial nonpositive curvature

We prove that two angle-compatible Coxeter generating sets of a given finitely generated Coxeter group are conjugate provided one of them does not admit any elementary twist. This confirms a basic case of a general conjecture which describes a potential solution to the isomorphism problem for Coxeter groups.