960 research outputs found
Conjugacy Growth and Conjugacy Width of Certain Branch Groups
The conjugacy growth function counts the number of distinct conjugacy classes
in a ball of radius . We give a lower bound for the conjugacy growth of
certain branch groups, among them the Grigorchuk group. This bound is a
function of intermediate growth. We further proof that certain branch groups
have the property that every element can be expressed as a product of uniformly
boundedly many conjugates of the generators. We call this property bounded
conjugacy width. We also show how bounded conjugacy width relates to other
algebraic properties of groups and apply these results to study the palindromic
width of some branch groups.Comment: Final version, to appear in IJA
Cancelation norm and the geometry of biinvariant word metrics
We study biinvariant word metrics on groups. We provide an efficient
algorithm for computing the biinvariant word norm on a finitely generated free
group and we construct an isometric embedding of a locally compact tree into
the biinvariant Cayley graph of a nonabelian free group. We investigate the
geometry of cyclic subgroups. We observe that in many classes of groups cyclic
subgroups are either bounded or detected by homogeneous quasimorphisms. We call
this property the bq-dichotomy and we prove it for many classes of groups of
geometric origin.Comment: 32 pages, to appear in Glasgow Journal of Mathematic
On the finiteness of the classifying space for the family of virtually cyclic subgroups
Given a group G, we consider its classifying space for the family of virtually cyclic subgroups. We show for many groups, including for example, one-relator groups, acylindrically hyperbolic groups, 3-manifold groups and CAT(0) cube groups, that they do not admit a finite model for this classifying space unless they are virtually cyclic. This settles a conjecture due to Juan-Pineda and Leary for these classes of groups
On the finiteness of the classifying space for the family of virtually cyclic subgroups
Given a group G, we consider its classifying space for the family of
virtually cyclic subgroups. We show for many groups, including for example,
one-relator groups, acylindrically hyperbolic groups, 3-manifold groups and
CAT(0) cube groups, that they do not admit a finite model for this classifying
space unless they are virtually cyclic. This settles a conjecture due to
Juan-Pineda and Leary for these classes of groups.Comment: Minor changes, to appear in Groups, Geometry, and Dynamic
Finitely generated infinite simple groups of infinite square width and vanishing stable commutator length
It is shown that there exist finitely generated infinite simple groups of
infinite commutator width and infinite square width on which there exists no
stably unbounded conjugation-invariant norm, and in particular stable
commutator length vanishes. Moreover, a recursive presentation of such a group
with decidable word and conjugacy problems is constructed.Comment: v4: 41 pages, 6 figures rescaled at 120%; references updated, typos
corrected, other minor corrections. v3: minor changes to the title, text and
figures. v2: 41 pages, 6 figures; correction: Ore's conjecture was proved in
2008; 2 references added. v1: 40 pages, 6 figure
Statistics and compression of scl
We obtain sharp estimates on the growth rate of stable commutator length on
random (geodesic) words, and on random walks, in hyperbolic groups and groups
acting nondegenerately on hyperbolic spaces. In either case, we show that with
high probability stable commutator length of an element of length is of
order .
This establishes quantitative refinements of qualitative results of
Bestvina-Fujiwara and others on the infinite dimensionality of 2-dimensional
bounded cohomology in groups acting suitably on hyperbolic spaces, in the sense
that we can control the geometry of the unit balls in these normed vector
spaces (or rather, in random subspaces of their normed duals).
As a corollary of our methods, we show that an element obtained by random
walk of length in a mapping class group cannot be written as a product of
fewer than reducible elements, with probability going to 1 as
goes to infinity. We also show that the translation length on the complex
of free factors of a random walk of length on the outer automorphism group
of a free group grows linearly in .Comment: Minor edits arising from referee's comments; 45 page
Stable commutator length in word-hyperbolic groups
In this paper we obtain uniform positive lower bounds on stable commutator
length in word-hyperbolic groups and certain groups acting on hyperbolic spaces
(namely the mapping class group acting on the complex of curves, and an
amalgamated free product acting on the Bass-Serre tree). If G is a word
hyperbolic group which is delta hyperbolic with respect to a symmetric
generating set S, then there is a positive constant C depending only on delta
and on |S| such that every element of G either has a power which is conjugate
to its inverse, or else the stable commutator length is at least equal to C. By
Bavard's theorem, these lower bounds on stable commutator length imply the
existence of quasimorphisms with uniform control on the defects; however, we
show how to construct such quasimorphisms directly.
We also prove various separation theorems, constructing homogeneous
quasimorphisms (again with uniform estimates) which are positive on some
prescribed element while vanishing on some family of independent elements whose
translation lengths are uniformly bounded.
Finally, we prove that the first accumulation point for stable commutator
length in a torsion-free word hyperbolic group is contained between 1/12 and
1/2. This gives a universal sense of what it means for a conjugacy class in a
hyperbolic group to have a small stable commutator length, and can be thought
of as a kind of "homological Margulis lemma".Comment: 27 pages, 1 figures; version 4: incorporates referee's suggestion
Filtrations and Distortion in Infinite-Dimensional Algebras
A tame filtration of an algebra is defined by the growth of its terms, which
has to be majorated by an exponential function. A particular case is the degree
filtration used in the definition of the growth of finitely generated algebras.
The notion of tame filtration is useful in the study of possible distortion of
degrees of elements when one algebra is embedded as a subalgebra in another. A
geometric analogue is the distortion of the (Riemannian) metric of a (Lie)
subgroup when compared to the metric induced from the ambient (Lie) group. The
distortion of a subalgebra in an algebra also reflects the degree of complexity
of the membership problem for the elements of this algebra in this subalgebra.
One of our goals here is to investigate, mostly in the case of associative or
Lie algebras, if a tame filtration of an algebra can be induced from the degree
filtration of a larger algebra
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