Mathematics

We show that the Steinberg group St( C 2 ,Z) associated with the Lie type C 2 and with integer coefficients can be realized as a quotient of the braid group B 6 by one relation. As an application we give a new braid-like presentation of the symplectic modular group Sp 4 (Z) .

We give a new characterisation of virtually free groups using graph minors. Namely, we prove that a finitely generated, infinite group is virtually free if and only if for any finite generating set, the corresponding Cayley graph is minor excluded. This answers a question of Ostrovskii and Rosenthal. The proof relies on showing that a finitely generated group that is minor excluded with respect to every finite generating set is accessible, using a graph-theoretic characterisation of accessibility due to Thomassen and Woess.

In this paper, we obtain a characterization of GVZ-groups in terms of commutators and monolithic quotients. This characterization is based on counting formulas due to Gallagher.

The knapsack problem for groups was introduced by Miasnikov, Nikolaev, and Ushakov. It is defined for each finitely generated group G and takes as input group elements g 1 ,?? g n ,g?�G and asks whether there are x 1 ,?? x n ?? with g x 1 1 ??g x n n =g . We study the knapsack problem for wreath products G?�H of groups G and H . Our main result is a characterization of those wreath products G?�H for which the knapsack problem is decidable. The characterization is in terms of decidability properties of the indiviual factors G and H . To this end, we introduce two decision problems, the intersection knapsack problem and its restriction, the positive intersection knapsack problem. Moreover, we apply our main result to H 3 (Z) , the discrete Heisenberg group, and to Baumslag-Solitar groups BS(1,q) for q?? . First, we show that the knapsack problem is undecidable for G?� H 3 (Z) for any G?? . This implies that for G?? and for infinite and virtually nilpotent groups H , the knapsack problem for G?�H is decidable if and only if H is virtually abelian and solvability of systems of exponent equations is decidable for G . Second, we show that the knapsack problem is decidable for G?�BS(1,q) if and only if solvability of systems of exponent equations is decidable for G .

We classify the locally compact second-countable (l.c.s.c.) groups A that are abelian and topologically characteristically simple. All such groups A occur as the monolith of some soluble l.c.s.c. group G of derived length at most 3 ; with known exceptions (specifically, when A is Q n or its dual for some n∈N ), we can take G to be compactly generated. This amounts to a classification of the possible isomorphism types of abelian chief factors of l.c.s.c. groups, which is of particular interest for the theory of compactly generated locally compact groups.

The Baumslag-Solitar group BS(2,3) , is a so-called non-Hopfian group, meaning that it has an epimorphism ϕ onto itself, that is not injective. In particular this is equivalent to saying that BS(2,3) has a non-trivial quotient that is isomorphic to itself. As a consequence the Cayley graph of BS(2,3) has a quotient that is isomorphic to itself up to change of generators. We describe this quotient on the graph-level and take a closer look at the most common epimorphism ϕ . We show its kernel is a free group of infinite rank with an explicit set of generators. Finally we show how ϕ appears as a morphism on fundamental groups induced by some continuous map. This point of view was communicated to the author by Gilbert Levitt.

A subgroup H of a group G is confined if the G -orbit of H under conjugation is bounded away from the trivial subgroup in the space Sub(G) of subgroups of G . We prove a commutator lemma for confined subgroups. For groups of homeomorphisms, this provides the exact analogue for confined subgroups (hence in particular for URSs) of the classical commutator lemma for normal subgroups: if G is a group of homeomorphisms of a Hausdorff space X and H is a confined subgroup of G , then H contains the derived subgroup of the rigid stabilizer of some open subset of X . We apply this commutator lemma to the study of URSs and actions on compact spaces of groups acting on rooted trees. We prove a theorem describing the structure of URSs of weakly branch groups and of their non-topologically free minimal actions. Among the applications of these results, we show: 1) if G is a finitely generated branch group, the G -action on ∂T has the smallest possible orbital growth among all faithful G -actions; 2) if G is a finitely generated branch group, then every embedding from G into a group of homeomorphisms of strongly bounded type (e.g. a bounded automaton group) must be spatially realized; 3) if G is a finitely generated weakly branch group, then G does not embed into the group IET of interval exchange transformations.

The unit conjecture, commonly attributed to Kaplansky, predicts that if K is a field and G is a torsion-free group then the only units of the group ring K[G] are the trivial units, that is, the non-zero scalar multiples of group elements. We give a concrete counterexample to this conjecture; the group is virtually abelian and the field is order two.

We construct a finitely generated 2-dimensional group that acts properly on a locally finite CAT(0) cube complex but does not act properly on a finite dimensional CAT(0) cube complex.

We prove the analogue of Johannson's Deformation Theorem for PD3 pairs.