Centroids and the Rapid Decay property in mapping class groups

We study a notion of a Lipschitz, permutation-invariant "centroid" for triples of points in mapping class groups MCG(S), which satisfies a certain polynomial growth bound. A consequence (via work of Drutu-Sapir or Chatterji-Ruane) is the Rapid Decay Property for MCG(S).Comment: v3. Numerous typos fixed and some arguments elucidate

The Rapid Decay property and centroids in groups

This is a survey of methods of proving or disproving the Rapid Decay property in groups. We present a centroid property of group actions on metric spaces. That property is a generalized (and corrected) version of the property (**)-relative hyperbolicity" from our paper with Cornelia Drutu, math/0405500, and implies the Rapid Decay (RD) property. We show that several properties which are known to imply RD also imply the centroid property. Thus uniform lattices in many semi-simple Lie groups, Artin groups of large type and the mapping class groups have the centroid property. We also present a simple "non-amenability-like" property that follows from RD, and give an easy example of a group without RD and without any amenable subgroup with superpolynomial growth, some misprints in other sections are corrected.Comment: 27 pages; v2: Section 2 corrected, added a reference to Olshansii's preprint arXiv:1406.0336 in Section 4. v3: Section 3.6 on graph products of groups is added. v3: Small correction in the proof of Theorem 2.3, estimate slightly improve

On the degree of rapid decay

A finitely generated group \G equipped with a word-length is said to satisfy property RD if there are $C, s\geq 0$ such that, for all non-negative integers $n$, we have $\|a\|\leq C (1+n)^s \|a\|_2$ whenever a\in\C\G is supported on elements of length at most $n$. We show that, for infinite \G, the degree $s$ is at least 1/2.Comment: 6 pages, final versio

Higher rank lattices are not coarse median

We show that symmetric spaces and thick affine buildings which are not of spherical type $A_1^r$ have no coarse median in the sense of Bowditch. As a consequence, they are not quasi-isometric to a CAT(0) cube complex, answering a question of Haglund. Another consequence is that any lattice in a simple higher rank group over a local field is not coarse median.Comment: 13 pages, 2 figures. To appear in Algebraic & Geometric Topolog

The 4-string Braid group B4B_4 has property RD and exponential mesoscopic rank

We prove that the braid group $B_4$ on 4 strings, as well as its central quotient $B_4/$, have the property RD of Haagerup-Jolissaint. It follows that the automorphism group \Aut(F_2) of the free group $F_2$ on 2 generators has property RD. We also prove that the braid group $B_4$ is a group of intermediate rank (of dimension 3). Namely, we show that both $B_4$ and its central quotient have exponential mesoscopic rank, i.e., that they contain exponentially many large flat balls which are not included in flats.Comment: reference added, minor correction

Discovering human activities from binary data in smart homes

With the rapid development in sensing technology, data mining, and machine learning fields for human health monitoring, it became possible to enable monitoring of personal motion and vital signs in a manner that minimizes the disruption of an individual’s daily routine and assist individuals with difficulties to live independently at home. A primary difficulty that researchers confront is acquiring an adequate amount of labeled data for model training and validation purposes. Therefore, activity discovery handles the problem that activity labels are not available using approaches based on sequence mining and clustering. In this paper, we introduce an unsupervised method for discovering activities from a network of motion detectors in a smart home setting. First, we present an intra-day clustering algorithm to find frequent sequential patterns within a day. As a second step, we present an inter-day clustering algorithm to find the common frequent patterns between days. Furthermore, we refine the patterns to have more compressed and defined cluster characterizations. Finally, we track the occurrences of various regular routines to monitor the functional health in an individual’s patterns and lifestyle. We evaluate our methods on two public data sets captured in real-life settings from two apartments during seven-month and three-month periods

Geometry of infinitely presented small cancellation groups, Rapid Decay and quasi-homomorphisms

We study the geometry of infinitely presented groups satisfying the small cancelation condition C'(1/8), and define a standard decomposition (called the criss-cross decomposition) for the elements of such groups. We use it to prove the Rapid Decay property for groups with the stronger small cancelation property C'(1/10). As a consequence, the Metric Approximation Property holds for the reduced C*-algebra and for the Fourier algebra of such groups. Our method further implies that the kernel of the comparison map between the bounded and the usual group cohomology in degree 2 has a basis of power continuum. The present work can be viewed as a first non-trivial step towards a systematic investigation of direct limits of hyperbolic groups.Comment: 40 pages, 8 figure

Invariance of coarse median spaces under relative hyperbolicity

We show that, for finitely generated groups, the property of admitting a coarse median structure is preserved under relative hyperbolicity