55,894 research outputs found
Poincar\'e profiles of groups and spaces
We introduce a spectrum of monotone coarse invariants for metric measure
spaces called Poincar\'{e} profiles. The two extremes of this spectrum
determine the growth of the space, and the separation profile as defined by
Benjamini--Schramm--Tim\'{a}r. In this paper we focus on properties of the
Poincar\'{e} profiles of groups with polynomial growth, and of hyperbolic
spaces, where we deduce a connection between these profiles and conformal
dimension. As applications, we use these invariants to show the non-existence
of coarse embeddings in a variety of examples.Comment: 55 pages. To appear in Revista Matem\'atica Iberoamerican
Volume distortion in groups
Given a space in , a cycle in may be filled with a chain in two
ways: either by restricting the chain to or by allowing it to be anywhere
in . When the pair acts on , we define the -volume
distortion function of in to measure the large-scale difference between
the volumes of such fillings. We show that these functions are quasi-isometry
invariants, and thus independent of the choice of spaces, and provide several
bounds in terms of other group properties, such as Dehn functions. We also
compute the volume distortion in a number of examples, including characterizing
the -volume distortion of in , where is a
diagonalizable matrix. We use this to prove a conjecture of Gersten.Comment: 27 pages, 10 figure
The geometry of flip graphs and mapping class groups
The space of topological decompositions into triangulations of a surface has
a natural graph structure where two triangulations share an edge if they are
related by a so-called flip. This space is a sort of combinatorial
Teichm\"uller space and is quasi-isometric to the underlying mapping class
group. We study this space in two main directions. We first show that strata
corresponding to triangulations containing a same multiarc are strongly convex
within the whole space and use this result to deduce properties about the
mapping class group. We then focus on the quotient of this space by the mapping
class group to obtain a type of combinatorial moduli space. In particular, we
are able to identity how the diameters of the resulting spaces grow in terms of
the complexity of the underlying surfaces.Comment: 46 pages, 23 figure
The classification of punctured-torus groups
Thurston's ending lamination conjecture proposes that a finitely generated
Kleinian group is uniquely determined (up to isometry) by the topology of its
quotient and a list of invariants that describe the asymptotic geometry of its
ends. We present a proof of this conjecture for punctured-torus groups. These
are free two-generator Kleinian groups with parabolic commutator, which should
be thought of as representations of the fundamental group of a punctured torus.
As a consequence we verify the conjectural topological description of the
deformation space of punctured-torus groups (including Bers' conjecture that
the quasi-Fuchsian groups are dense in this space) and prove a rigidity
theorem: two punctured-torus groups are quasi-conformally conjugate if and only
if they are topologically conjugate.Comment: 67 pages, published versio
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.
Conformal dimension via subcomplexes for small cancellation and random groups
We find new bounds on the conformal dimension of small cancellation groups.
These are used to show that a random few relator group has conformal dimension
2+o(1) asymptotically almost surely (a.a.s.). In fact, if the number of
relators grows like l^K in the length l of the relators, then a.a.s. such a
random group has conformal dimension 2+K+o(1). In Gromov's density model, a
random group at density d<1/8 a.a.s. has conformal dimension .
The upper bound for C'(1/8) groups has two main ingredients:
-cohomology (following Bourdon-Kleiner), and walls in the Cayley
complex (building on Wise and Ollivier-Wise). To find lower bounds we refine
the methods of [Mackay, 2012] to create larger `round trees' in the Cayley
complex of such groups.
As a corollary, in the density model at d<1/8, the density d is determined,
up to a power, by the conformal dimension of the boundary and the Euler
characteristic of the group.Comment: v1: 42 pages, 21 figures; v2: 44 pages, 20 figures. Improved
exposition, final versio
Equality of Lifshitz and van Hove exponents on amenable Cayley graphs
We study the low energy asymptotics of periodic and random Laplace operators
on Cayley graphs of amenable, finitely generated groups. For the periodic
operator the asymptotics is characterised by the van Hove exponent or zeroth
Novikov-Shubin invariant. The random model we consider is given in terms of an
adjacency Laplacian on site or edge percolation subgraphs of the Cayley graph.
The asymptotic behaviour of the spectral distribution is exponential,
characterised by the Lifshitz exponent. We show that for the adjacency
Laplacian the two invariants/exponents coincide. The result holds also for more
general symmetric transition operators. For combinatorial Laplacians one has a
different universal behaviour of the low energy asymptotics of the spectral
distribution function, which can be actually established on quasi-transitive
graphs without an amenability assumption. The latter result holds also for long
range bond percolation models
Conformal dimension and random groups
We give a lower and an upper bound for the conformal dimension of the
boundaries of certain small cancellation groups. We apply these bounds to the
few relator and density models for random groups. This gives generic bounds of
the following form, where is the relator length, going to infinity.
(a) 1 + 1/C < \Cdim(\bdry G) < C l / \log(l), for the few relator model,
and
(b) 1 + l / (C\log(l)) < \Cdim(\bdry G) < C l, for the density model, at
densities .
In particular, for the density model at densities , as the relator
length goes to infinity, the random groups will pass through infinitely
many different quasi-isometry classes.Comment: 32 pages, 4 figures. v2: Final version. Main result improved to
density < 1/16. Many minor improvements. To appear in GAF
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