A group is properly 3-realizable if it is the fundamental group of a compact polyhedron whose universal covering is proper homotopically equivalent to some topological 3-manifold. We proved that when such a group is also quasi-simply filtered then it has pro-(finitely generated free) fundamental group at infinity and semi-stable ends. Conjecturally the quasi-simply filtration assumption is superfluous. Using these restrictions we provided the first examples of finitely presented groups which are not properly 3-realizable, for instance large families of Coxeter groups.

COBISS.SI-ID: 16297817

For a compact right-angled polyhedron R in Lobachevskii space {\mathbb H}^3, let {\rm{vol}}(R) denote its volume and {\rm{vert}}(R), the number of its vertices. Upper and lower bounds for {\rm{vol}}(R) were recently obtained by Atkinson in terms of {\rm{vert}}(R). By constructing a two-parameter family of polyhedra, we show that the asymptotic upper bound 5v_3/8, where v_3 is the volume of the ideal regular tetrahedron in {\mathbb H}^3, is a double limit point for the ratios {\rm{vol}}(R)/{\rm{vert}}(R). Moreover, we improve the lower bound in the case {\rm{vert}}(R)\leqslant 56.

COBISS.SI-ID: 15843929

We investigate the concordance properties of "parallel links" P(K), given by the (2,0) cable of a knot K. We focus on the question: if P(K) is concordant to a split link, is K necessarily slice? We show that if P(K) is smoothly concordant to a split link, then many smooth concordance invariants of K must vanish, including the \tau and s-invariants, as well as suitably normalized d-invariants of Dehn surgeries on K. We also investigate the (2,2\ell) cables P_{\ell}(K), and find obstructions to smooth concordance to the sum of the (2,2\ell) torus link and a split link.

COBISS.SI-ID: 16946265

It has been an open question for a long time whether every countable group can be realized as a fundamental group of a compact metric space. Such realizations are not hard to obtain for compact or metric spaces but the combination of both properties turn out to be quite restrictive for the fundamental group. The problem has been studied by many topologists (including Cannon and Conner) but the solution has not been found. In this paper we prove that any countable group can be realized as the fundamental group of a compact subspace of {\mathbb R}^4. According to the theorem of Shelah [S. Shelah, Proc. Amer. Math. Soc. 103, no. 2, (1988), 627-632] such space cannot be locally path connected if the group is not finitely generated. The theorem is proved by an explicit construction of an appropriate space X_G for every countable group G.

COBISS.SI-ID: 16654681

Recent research in coarse geometry revealed similarities between certain concepts of analysis, large scale geometry, and topology. Property A of Yu is the coarse analog of amenability for groups and its generalization (exact spaces) was later strengthened to be the large scale analog of paracompact spaces using partitions of unity. In this paper we go deeper into divulging analogies between coarse amenability and paracompactness. In particular, we define a new coarse analog of paracompactness modeled on the defining characteristics of expanders. That analog gives an easy proof of three categories of spaces being coarsely non-amenable: expander sequences, graph spaces with girth approaching infinity, and unions of powers of a finite nontrivial group.

COBISS.SI-ID: 16935513