3,482 research outputs found
Assouad dimension of self-affine carpets
We calculate the Assouad dimension of the self-affine carpets of Bedford and
McMullen, and of Lalley and Gatzouras. We also calculate the conformal Assouad
dimension of those carpets that are not self-similar.Comment: 10 pages, 3 figure
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
Differentiable structures on metric measure spaces: A Primer
This is an exposition of the theory of differentiable structures on metric
measures spaces, in the sense of Cheeger and Keith.Comment: 23 page
Poorly connected groups
We investigate groups whose Cayley graphs have poor\-ly connected subgraphs.
We prove that a finitely generated group has bounded separation in the sense of
Benjamini--Schramm--Tim\'ar if and only if it is virtually free. We then prove
a gap theorem for connectivity of finitely presented groups, and prove that
there is no comparable theorem for all finitely generated groups. Finally, we
formulate a connectivity version of the conjecture that every group of type
with no Baumslag-Solitar subgroup is hyperbolic, and prove it for groups with
at most quadratic Dehn function.Comment: 14 pages. Changes to v2: Proof of the Theorem 1.2 shortened, Theorem
1.4 added completing the no-gap result outlined in v
Quasi-hyperbolic planes in relatively hyperbolic groups
We show that any group that is hyperbolic relative to virtually nilpotent
subgroups, and does not admit peripheral splittings, contains a
quasi-isometrically embedded copy of the hyperbolic plane. In natural
situations, the specific embeddings we find remain quasi-isometric embeddings
when composed with the inclusion map from the Cayley graph to the coned-off
graph, as well as when composed with the quotient map to "almost every"
peripheral (Dehn) filling.
We apply our theorem to study the same question for fundamental groups of
3-manifolds.
The key idea is to study quantitative geometric properties of the boundaries
of relatively hyperbolic groups, such as linear connectedness. In particular,
we prove a new existence result for quasi-arcs that avoid obstacles.Comment: v1: 32 pages, 4 figures. v2: 38 pages, 4 figures. v3: 44 pages, 4
figures. An application (Theorem 1.2) is weakened as there was an error in
its proof in section 7, all other changes minor, improved expositio
Balanced walls for random groups
We study a random group G in the Gromov density model and its Cayley complex
X. For density < 5/24 we define walls in X that give rise to a nontrivial
action of G on a CAT(0) cube complex. This extends a result of Ollivier and
Wise, whose walls could be used only for density < 1/5. The strategy employed
might be potentially extended in future to all densities < 1/4.Comment: 18 pages, 2 figures. v2: Minor improvements, final versio
A Metrizable Topology on the Contracting Boundary of a Group
The 'contracting boundary' of a proper geodesic metric space consists of
equivalence classes of geodesic rays that behave like rays in a hyperbolic
space. We introduce a geometrically relevant, quasi-isometry invariant topology
on the contracting boundary. When the space is the Cayley graph of a finitely
generated group we show that our new topology is metrizable.Comment: v1: 26 pages, 3 figures; v2: 44 pages, 6 figures, additional results;
v3: 46 pages, 7 figures, minor change
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
Embedding relatively hyperbolic groups in products of trees
We show that a relatively hyperbolic group quasi-isometrically embeds in a
product of finitely many trees if the peripheral subgroups do, and we provide
an estimate on the minimal number of trees needed. Applying our result to the
case of 3-manifolds, we show that fundamental groups of closed 3-manifolds have
linearly controlled asymptotic dimension at most 8. To complement this result,
we observe that fundamental groups of Haken 3-manifolds with non-empty boundary
have asymptotic dimension 2.Comment: v1: 18 pages; v2: 20 pages, minor change
The Epileptic Constitution: A Study of Seventy Cases of Epilepsy at Whittingham Mental Hospital, Preston, Lancs
Scheme of Thesis: I propose to record the results of observations made on a group of epileptic patients at present under my care at the County Mental Hospital, Whittingham, Preston. Anatomical, physiological and psychological animalies of epileptics are presented and discussed. Finally it is claimed that the facts emerging from this study indicate a constitutional basis in epilepsy
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