Francisco González-Acuña

Cohopficity of 3-manifold groups

Abstract A group G is said to be cohopfian, if every monomorphism from G to itself is an automorphism. In this paper, we first give necessary and sufficient conditions for the group of a closed, virtually geometric 3-manifold containing no two-sided projective planes to be cohopfian. Next, we show that a finitely generated, proper, free product of groups has a proper subgroup of finite index isomorphic to itself if and only if the original group is the free product of a group of order two with itself. Using this result, we then show that, if a closed 3-manifold N contains at least one two-sided projective plane and is not a homotopy projective plane cross a circle, then the group of N does not imbed in itself with finite index (greater than one). Finally, we give an example to show that this last result is false when the phrase, “with finite index,” is omitted.

Jørgensen subgroups of the Picard group

Let G be a subgroup of rank two of the Mobius group PSL(2, C). The Jorgensen number J (G) of G is defined by J (G) = inf tr 2 A 4 + tr[ A, B] 2 : A, B = G . We describ e all subgroups G of the Picard group PSL(2, Z + iZ) with J (G) = 1.

SELF-INTERSECTION NUMBERS OF PATHS IN COMPACT SURFACES

In this paper, we give an algorithm to calculate the minimal self-intersection number of paths in a compact surface with boundary representing a given element of the free group F(x1, x2, …, xn). In particular, this algorithm says whether or not a word in x1, x2, …, xn is representable by a simple path. Our algorithm is simpler than similar algorithms given previously. In the case of a disk with n holes the problem is equivalent to the problem of deciding which relators can appear in an Artin n-presentation.

Unsolvable problems about higher-dimensional knots and related groups

We consider classes of fundamental groups of complements of various kinds of codimension 2 embeddings and show that, in general, the problem of deciding whether or not a group in one class belongs to a smaller class is algorithmically unsolvable.

A KNOT-THEORETIC EQUIVALENT OF THE KERVAIRE CONJECTURE

Kervaires conjecture is equivalent to the following one: If F is a compact orientable nonseparating surface properly embedded in a knot exterior E then π1(E/F) ≈ Z.