25 research outputs found
The asymptotic density of finite-order elements in virtually nilpotent groups
Let G be a finitely generated group with a given word metric. The asymptotic
density of elements in G that have a particular property P is defined to be the
limit, as r goes to infinity, of the proportion of elements in the ball of
radius r which have the property P. We obtain a formula to compute the
asymptotic density of finite-order elements in any virtually nilpotent group.
Further, we show that the spectrum of numbers that occur as such asymptotic
densities consists of exactly the rational numbers in [0,1).Comment: 26 page
Divergence in right-angled Coxeter groups
Let W be a 2-dimensional right-angled Coxeter group. We characterise such W
with linear and quadratic divergence, and construct right-angled Coxeter groups
with divergence polynomial of arbitrary degree. Our proofs use the structure of
walls in the Davis complex.Comment: This version incorporates the referee's comments. It contains the
complete appendix (which will be abbreviated in the journal version). To
appear in Transactions of the AM
Divergence in right-angled Coxeter groups
Let W be a 2-dimensional right-angled Coxeter group. We characterise such W with linear and quadratic divergence, and construct right-angled Coxeter groups with divergence polynomial of arbitrary degree. Our proofs use the structure of walls in the Davis complex
Morse theory and conjugacy classes of finite subgroups II
We construct a hyperbolic group with a finitely presented subgroup, which has
infinitely many conjugacy classes of finite-order elements.
We also use a version of Morse theory with high dimensional horizontal cells
and use handle cancellation arguments to produce other examples of subgroups of
CAT(0) groups with infinitely many conjugacy classes of finite-order elements.Comment: 18 pages, 7 figure
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
Super-exponential distortion of subgroups of CAT(-1) groups
We construct 2-dimensional CAT(-1) groups which contain free subgroups with
arbitrary iterated exponential distortion, and with distortion higher than any
iterated exponential.Comment: 6 pages, 3 figure
Bowditch\u27s JSJ tree and the quasi-isometry classification of certain Coxeter groups
Bowditch\u27s JSJ tree for splittings over 2-ended subgroups is a quasi-isometry invariant for 1-ended hyperbolic groups which are not cocompact Fuchsian [Bowditch, Acta Math. 180 (1998) 145-186]. Our main result gives an explicit, computable \u27visual\u27 construction of this tree for certain hyperbolic right-angled Coxeter groups. As an application of our construction we identify a large class of such groups for which the JSJ tree, and hence the visual boundary, is a complete quasi-isometry invariant, and thus the quasi-isometry problem is decidable. We also give a direct proof of the fact that among the Coxeter groups we consider, the cocompact Fuchsian groups form a rigid quasi-isometry class. In Appendix B, written jointly with Christopher Cashen, we show that the JSJ tree is not a complete quasi-isometry invariant for the entire class of Coxeter groups we consider
Filling loops at infinity in the mapping class group
We study the Dehn function at infinity in the mapping class group, finding a
polynomial upper bound of degree four. This is the same upper bound that holds
for arbitrary right-angled Artin groups.Comment: 7 pages, 2 figures; this note presents a result which was contained
in an earlier version of "Pushing fillings in right-angled Artin groups"
(arXiv:1004.4253) but is independent of the techniques in that pape
Pushing fillings in right-angled Artin groups
We construct "pushing maps" on the cube complexes that model right-angled
Artin groups (RAAGs) in order to study filling problems in certain subsets of
these cube complexes. We use radial pushing to obtain upper bounds on higher
divergence functions, finding that the k-dimensional divergence of a RAAG is
bounded by r^{2k+2}. These divergence functions, previously defined for
Hadamard manifolds to measure isoperimetric properties "at infinity," are
defined here as a family of quasi-isometry invariants of groups; thus, these
results give new information about the QI classification of RAAGs. By pushing
along the height gradient, we also show that the k-th order Dehn function of a
Bestvina-Brady group is bounded by V^{(2k+2)/k}. We construct a class of RAAGs
called "orthoplex groups" which show that each of these upper bounds is sharp.Comment: The result on the Dehn function at infinity in mapping class groups
has been moved to the note "Filling loops at infinity in the mapping class
group.