Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userAutomorphisms of the k-Curve Graph
72
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
Given a natural number k and an orientable surface S of finite type, define the k-curve graph to be the graph with vertices corresponding to isotopy classes of essential simple closed curves on S and with edges corresponding to pairs of such curves admitting representatives that intersect at most k times. We prove that the automorphism group of the k-curve graph of a surface S is isomorphic to the extended mapping class group for all k sufficiently small with respect to the Euler characteristic of S. We prove the same result for the so-called systolic complex, a variant of the curve graph whose complete subgraphs encode the intersection patterns for any collection of systoles with respect to a hyperbolic metric. This resolves a conjecture of Schmutz Schaller.
rate research
Read More
Very sparse random graphs are known to typically be singular (i.e., have singular adjacency matrix), due to the presence of low-degree dependencies such as isolated vertices and pairs of degree-1 vertices with the same neighbourhood. We prove that these kinds of dependencies are in some sense the only causes of singularity: for constants $kge 3$ and $lambda > 0$, an ErdH os--Renyi random graph $Gsimmathbb{G}(n,lambda/n)$ with $n$ vertices and edge probability $lambda/n$ typically has the property that its $k$-core (its largest subgraph with minimum degree at least $k$) is nonsingular. This resolves a conjecture of Vu from the 2014 International Congress of Mathematicians, and adds to a short list of known nonsingularity theorems for extremely sparse random matrices with density $O(1/n)$. A key aspect of our proof is a technique to extract high-degree vertices and use them to boost the rank, starting from approximate rank bounds obtainable from (non-quantitative) spectral convergence machinery due to Bordenave, Lelarge and Salez.
The paper establishes new relationship between cohomology, extensions and automorphisms of quandles. We derive a four term exact sequence relating quandle 1-cocycles, second quandle cohomology and certain group of automorphisms of an abelian extension of quandles. A non-abelian counterpart of this sequence involving dynamical cohomology classes is also established, and some applications to lifting of quandle automorphisms are given. Viewing the construction of the conjugation, the core and the generalised Alexander quandle of a group as an adjoint functor of some appropriate functor from the category of quandles to the category of groups, we prove that these functors map extensions of groups to extensions of quandles. Finally, we construct some natural group homomorphisms from the second cohomology of a group to the second cohomology of its core and conjugation quandles.
We determine the size of $k$-core in a large class of dense graph sequences. Let $G_n$ be a sequence of undirected, $n$-vertex graphs with edge weights ${a^n_{i,j}}_{i,j in [n]}$ that converges to a kernel $W:[0,1]^2to [0,+infty)$ in the cut metric. Keeping an edge $(i,j)$ of $G_n$ with probability $min { {a^n_{i,j}}/{n},1 }$ independently, we obtain a sequence of random graphs $G_n(frac{1}{n})$. Denote by $C_k(G)$ the size of $k$-core in graph $G$, by $X^W$ the branching process associated with the kernel $W$, by $mathcal{A}$ the property of a branching process that the initial particle has at least $k$ children, each of which has at least $k-1$ children, each of which has at least $k-1$ children, and so on. Using branching process and theory of dense graph limits, under mild assumptions we obtain the size of $k$-core of random graphs $G_n(frac{1}{n})$, begin{align*} C_kleft(G_nleft(frac{1}{n}right)right) =n mathbb{P}_{X^W}left(mathcal{A}right) +o_p(n). end{align*} Our result can also be used to obtain the threshold of appearance of a $k$-core of order $n$. In addition, we obtain a probabilistic result concerning cut-norm and branching process which might be of independent interest.
Mirzakhani wrote two papers on counting curves of given type on a surface: one for simple curves, and one for arbitrary ones. We give a complete argument deriving Mirzakhanis result for general curves from the one about simple ones. We then sketch an argument to give a new proof of both results -- full details and other related matters will appear in a book we intend to write.
The fundamental group of the Menger universal curve is uncountable and not free, although all of its finitely generated subgroups are free. It contains an isomorphic copy of the fundamental group of every one-dimensional separable metric space and an isomorphic copy of the fundamental group of every planar Peano continuum. We give an explicit and systematic combinatorial description of the fundamental group of the Menger universal curve and its generalized Cayley graph in terms of word sequences. The word calculus, which requires only two letters and their inverses, is based on Pasynkovs partial topological product representation and can be expressed in terms of a variation on the classical puzzle known as the Towers of Hanoi.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
