123 research outputs found
On the isomorphism problem for generalized Baumslag-Solitar groups
Generalized Baumslag-Solitar groups (GBS groups) are groups that act on trees
with infinite cyclic edge and vertex stabilizers. Such an action is described
by a labeled graph (essentially, the quotient graph of groups). This paper
addresses the problem of determining whether two given labeled graphs define
isomorphic groups; this is the isomorphism problem for GBS groups. There are
two main results and some applications. First, we find necessary and sufficient
conditions for a GBS group to be represented by only finitely many reduced
labeled graphs. These conditions can be checked effectively from any labeled
graph. Then we show that the isomorphism problem is solvable for GBS groups
whose labeled graphs have first Betti number at most one.Comment: 30 pages. v2: 35 pages, 3 figures; minor revisions and reformattin
Whitehead moves for G-trees
We generalize the familiar notion of a Whitehead move from Culler and
Vogtmann's Outer space to the setting of deformation spaces of G-trees.
Specifically, we show that there are two moves, each of which transforms a
reduced G-tree into another reduced G-tree, that suffice to relate any two
reduced trees in the same deformation space. These two moves further factor
into three moves between reduced trees that have simple descriptions in terms
of graphs of groups. This result has several applications.Comment: v1: 9 pages; v2: 10 pages, minor revisions and one added referenc
Morse theory and conjugacy classes of finite subgroups
We construct a CAT(0) group containing a finitely presented subgroup with
infinitely many conjugacy classes of finite-order elements. Unlike previous
examples (which were based on right-angled Artin groups) our ambient CAT(0)
group does not contain any rank 3 free abelian subgroups.
We also construct examples of groups of type F_n inside mapping class groups,
Aut(F), and Out(F) which have infinitely many conjugacy classes of finite-order
elements.Comment: 10 pages, 4 figure
- …