A remark on the word length in surface groups

Let $\Sigma$ be a surface of negative Euler characteristic and $S$ a generating set for $\pi_1(\Sigma,p)$ consisting of simple loops that are pairwise disjoint (except at $p$). We show that the word length with respect to $S$ of an element of $\pi_1(\Sigma,p)$ is given by its intersection number with a well-chosen collection of curves and arcs on $\Sigma$. The same holds for the word length of (a free homotopy class of) an immersed curve on $\Sigma$. As a consequence, we obtain the asymptotic growth of the number of immersed curves of bounded word length, as the length grows, in each mapping class group orbit.Comment: 18 pages, 4 figures. Corollary 4.1 added in new versio

The Margulis region and screw parabolic elements of bounded type

Given a discrete subgroup of the isometries of n-dimensional hyperbolic space there is always a region kept precisely invariant under the stabilizer of a parabolic fixed point, called the Margulis region. While in dimensions 2 and 3 this region is a horoball, it has in general a more complicated shape due to the existence of screw parabolic elements in higher dimensions. In fact, P. Susskind has shown that in a discrete group acting on hyperbolic 4-space containing a screw parabolic element with irrational rotation, the corresponding Margulis region does not contain a horoball. In this paper we describe the asymptotic behavior of the boundary of the Margulis region when the irrational screw parabolic is of bounded type. As a corollary we show that the region is quasi-isometric to a horoball. Although Y. Kim has shown that two screw parabolic isometries with irrational rotation are not conjugate by any quasi-isometry of hyperbolic 4-space, this corollary implies that their corresponding Margulis regions (in the bounded type case) are quasi-isometric

On Margulis cusps of hyperbolic 4-manifolds

We study the geometry of the Margulis region associated with an irrational screw translation $g$ acting on the 4-dimensional real hyperbolic space. This is an invariant domain with the parabolic fixed point of $g$ on its boundary which plays the role of an invariant horoball for a translation in dimensions $\leq 3$. The boundary of the Margulis region is described in terms of a function $B_\alpha : [0,\infty) \to {\mathbb R}$ which solely depends on the rotation angle $\alpha \in {\mathbb R}/{\mathbb Z}$ of $g$. We obtain an asymptotically universal upper bound for $B_\alpha(r)$ as $r \to \infty$ for arbitrary irrational $\alpha$, as well as lower bounds when $\alpha$ is Diophatine and the optimal bound when $\alpha$ is of bounded type. We investigate the implications of these results for the geometry of Margulis cusps of hyperbolic 4-manifolds that correspond to irrational screw translations acting on the universal cover. Among other things, we prove bi-Lipschitz rigidity of these cusps.Comment: 34 pages, 6 figure

Counting Curves in Hyperbolic Surfaces

Let $\Sigma$ be a hyperbolic surface. We study the set of curves on $\Sigma$ of a given type, i.e. in the mapping class group orbit of some fixed but otherwise arbitrary $\gamma_0$. For example, in the particular case that $\Sigma$ is a once-punctured torus, we prove that the cardinality of the set of curves of type $\gamma_0$ and of at most length $L$ is asymptotic to $L^2$ times a constant.Comment: 49 pages, 11 (mostly hand-drawn) figure

Simplicial embeddings between multicurve graphs

We study some graphs associated to a surface, called k-multicurve graphs, which interpolate between the curve complex and the pants graph. Our main result is that, under certain conditions, simplicial embeddings between multicurve graphs are induced by $\pi_1$-injective embeddings of the corresponding surfaces. We also prove the rigidity of the multicurve graphs.Comment: New introduction and some changes in Section 2, main results unchanged. References added. 18 pages, 5 figure

The Effective Exchange Rate Index KIX - Theory and Practice

The world trade patterns have changed over the recent years, and for Sweden many important trading partners have emerged. The National Institute of Economic Research has therefore compiled a new effective exchange rate of index for the Swedish krona, KIX. KIX is a chain-linked index that includes the currencies of 32 countries. The weight attached to each country is based on the patterns of international trade in manufactured goods and commodities. The index includes a number of emerging economies and gives in this way a more adequate definition of the effective exchange rate of the krona than the traditional indices. Furthermore, it allows for the changes in relative importance of Sweden´s trading partners. The calculation of KIX is now documented in a Working Paper: The Effective Exchange Rate Index KIX - Theory and Practise.

Finding Influential Users in Social Media Using Association Rule Learning

Influential users play an important role in online social networks since users tend to have an impact on one other. Therefore, the proposed work analyzes users and their behavior in order to identify influential users and predict user participation. Normally, the success of a social media site is dependent on the activity level of the participating users. For both online social networking sites and individual users, it is of interest to find out if a topic will be interesting or not. In this article, we propose association learning to detect relationships between users. In order to verify the findings, several experiments were executed based on social network analysis, in which the most influential users identified from association rule learning were compared to the results from Degree Centrality and Page Rank Centrality. The results clearly indicate that it is possible to identify the most influential users using association rule learning. In addition, the results also indicate a lower execution time compared to state-of-the-art methods

Do we really need to catch them all? A new User-guided Social Media Crawling method

With the growing use of popular social media services like Facebook and Twitter it is challenging to collect all content from the networks without access to the core infrastructure or paying for it. Thus, if all content cannot be collected one must consider which data are of most importance. In this work we present a novel User-guided Social Media Crawling method (USMC) that is able to collect data from social media, utilizing the wisdom of the crowd to decide the order in which user generated content should be collected to cover as many user interactions as possible. USMC is validated by crawling 160 public Facebook pages, containing content from 368 million users including 1.3 billion interactions, and it is compared with two other crawling methods. The results show that it is possible to cover approximately 75% of the interactions on a Facebook page by sampling just 20% of its posts, and at the same time reduce the crawling time by 53%. In addition, the social network constructed from the 20% sample contains more than 75% of the users and edges compared to the social network created from all posts, and it has similar degree distribution