19,649 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
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
Stellar Differential Rotation and Coronal Timescales
We investigate the timescales of evolution of stellar coronae in response to
surface differential rotation and diffusion. To quantify this we study both the
formation time and lifetime of a magnetic flux rope in a decaying bipolar
active region. We apply a magnetic flux transport model to prescribe the
evolution of the stellar photospheric field, and use this to drive the
evolution of the coronal magnetic field via a magnetofrictional technique.
Increasing the differential rotation (i.e. decreasing the equator-pole lap
time) decreases the flux rope formation time. We find that the formation time
is dependent upon the geometric mean of the lap time and the surface diffusion
timescale. In contrast, the lifetime of flux ropes are proportional to the lap
time. With this, flux ropes on stars with a differential rotation of more than
eight times the solar value have a lifetime of less than two days. As a
consequence, we propose that features such as solar-like quiescent prominences
may not be easily observable on such stars, as the lifetimes of the flux ropes
which host the cool plasma are very short. We conclude that such high
differential rotation stars may have very dynamical coronae
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