49 research outputs found
Density of isoperimetric spectra
We show that the set of k-dimensional isoperimetric exponents of finitely
presented groups is dense in the interval [1, \infty) for k > 1. Hence there is
no higher-dimensional analogue of Gromov's gap (1,2) in the isoperimetric
spectrum.Comment: 34 pages, 3 figure
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
Veech surfaces and simple closed curves
We study the SL(2,R)-infimal lengths of simple closed curves on
half-translation surfaces. Our main result is a characterization of Veech
surfaces in terms of these lengths. We also revisit the "no small virtual
triangles" theorem of Smillie and Weiss and establish the following dichotomy:
the virtual triangle area spectrum of a half-translation surface either has a
gap above zero or is dense in a neighborhood of zero. These results make use of
the auxiliary polygon associated to a curve on a half-translation surface, as
introduced by Tang and Webb.Comment: 12 pages. v2: added proof of continuity of infimal length functions
on quadratic differential space; 16 pages, one figure; to appear in Israel J.
Mat
Deformation and rigidity of simplicial group actions on trees
We study a notion of deformation for simplicial trees with group actions
(G-trees). Here G is a fixed, arbitrary group. Two G-trees are related by a
deformation if there is a finite sequence of collapse and expansion moves
joining them. We show that this relation on the set of G-trees has several
characterizations, in terms of dynamics, coarse geometry, and length functions.
Next we study the deformation space of a fixed G-tree X. We show that if X is
`strongly slide-free' then it is the unique reduced tree in its deformation
space.
These methods allow us to extend the rigidity theorem of Bass and Lubotzky to
trees that are not locally finite. This yields a unique factorization theorem
for certain graphs of groups. We apply the theory to generalized
Baumslag-Solitar groups and show that many have canonical decompositions. We
also prove a quasi-isometric rigidity theorem for strongly slide-free G-trees.Comment: Published in Geometry and Topology at
http://www.maths.warwick.ac.uk/gt/GTVol6/paper8.abs.htm
Stable commutator length in Baumslag-Solitar groups and quasimorphisms for tree actions
This paper has two parts, on Baumslag-Solitar groups and on general G-trees.
In the first part we establish bounds for stable commutator length (scl) in
Baumslag-Solitar groups. For a certain class of elements, we further show that
scl is computable and takes rational values. We also determine exactly which of
these elements admit extremal surfaces.
In the second part we establish a universal lower bound of 1/12 for scl of
suitable elements of any group acting on a tree. This is achieved by
constructing efficient quasimorphisms. Calculations in the group BS(2,3) show
that this is the best possible universal bound, thus answering a question of
Calegari and Fujiwara. We also establish scl bounds for acylindrical tree
actions.
Returning to Baumslag-Solitar groups, we show that their scl spectra have a
uniform gap: no element has scl in the interval (0, 1/12).Comment: v2: minor changes, incorporates referee suggestions; v1: 36 pages, 10
figure
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
On stable commutator length of non-filling curves in surfaces
We give a new proof of rationality of stable commutator length (scl) of
certain elements in surface groups: those represented by curves that do not
fill the surface. Such elements always admit extremal surfaces for scl. These
results also hold more generally for non-filling 1-chains.Comment: 17 pages; three figures have been added, along with some minor edit