Proof of a conjecture on hadamard 2-groups

AbstractBy expanding on the results of James Davis, we prove by construction that every abelian 2-group that meets the exponent bound has a difference set

Hadamard 2-(63,31,15) designs invariant under the dihedral group of order 10

AbstractAll Hadamard 2-(63,31,15) designs invariant under the dihedral group of order 10 are constructed and classified up to isomorphism together with related Hadamard matrices of order 64. Affine 2-(64,16,5) designs can be obtained from Hadamard 2-(63,31,15) designs having line spreads by Rahilly’s construction [A. Rahilly, On the line structure of designs, Discrete Math. 92 (1991) 291–303]. The parameter set 2-(64,16,5) is one of two known sets when there exists several nonisomorphic designs with the same parameters and p-rank as the design obtained from the points and subspaces of a given dimension in affine geometry AG(n,pm) (p a prime). It is established that an affine 2-(64,16,5) design of 2-rank 16 that is associated with a Hadamard 2-(63,31,15) design invariant under the dihedral group of order 10 is either isomorphic to the classical design of the points and hyperplanes in AG(3,4), or is one of the two exceptional designs found by Harada, Lam and Tonchev [M. Harada, C. Lam, V.D. Tonchev, Symmetric (4, 4)-nets and generalized Hadamard matrices over groups of order 4, Designs Codes Cryptogr. 34 (2005) 71–87]

Pushing fillings in right-angled Artin groups

We construct "pushing maps" on the cube complexes that model right-angled Artin groups (RAAGs) in order to study filling problems in certain subsets of these cube complexes. We use radial pushing to obtain upper bounds on higher divergence functions, finding that the k-dimensional divergence of a RAAG is bounded by r^{2k+2}. These divergence functions, previously defined for Hadamard manifolds to measure isoperimetric properties "at infinity," are defined here as a family of quasi-isometry invariants of groups; thus, these results give new information about the QI classification of RAAGs. By pushing along the height gradient, we also show that the k-th order Dehn function of a Bestvina-Brady group is bounded by V^{(2k+2)/k}. We construct a class of RAAGs called "orthoplex groups" which show that each of these upper bounds is sharp.Comment: The result on the Dehn function at infinity in mapping class groups has been moved to the note "Filling loops at infinity in the mapping class group.

Asymptotic Geometry in the product of Hadamard spaces with rank one isometries

In this article we study asymptotic properties of certain discrete groups $\Gamma$ acting by isometries on a product \XX=\XX_1\times \XX_2 of locally compact Hadamard spaces. The motivation comes from the fact that Kac-Moody groups over finite fields, which can be seen as generalizations of arithmetic groups over function fields, belong to this class of groups. Hence one may ask whether classical properties of discrete subgroups of higher rank Lie groups as in [MR1437472] and [MR1933790] hold in this context. In the first part of the paper we describe the structure of the geometric limit set of $\Gamma$ and prove statements analogous to the results of Benoist in [MR1437472]. The second part is concerned with the exponential growth rate $\delta_\theta(\Gamma)$ of orbit points in \XX with a prescribed so-called "slope" $\theta\in (0,\pi/2)$, which appropriately generalizes the critical exponent in higher rank. In analogy to Quint's result in [MR1933790] we show that the homogeneous extension $\Psi_\Gamma$ to \RR_{\ge 0}^2 of $\delta_\theta(\Gamma)$ as a function of $\theta$ is upper semi-continuous and concave.Comment: 27 pages, to appear in Geometry & Topolog

Pushing fillings in right-angled Artin groups

We obtain bounds on the higher divergence functions of right-angled Artin groups (RAAGs), finding that the k-dimensional divergence of a RAAG is bounded above by r2k+2. These divergence functions, previously defined for Hadamard manifolds to measure isoperimetric properties at infinity, are defined here as a family of quasi-isometry invariants of groups. We also show that the kth order Dehn function of a Bestvina-Brady group is bounded above by V (2k+2)/k and construct a class of RAAGs called orthoplex groups which show that each of these upper bounds is sharp. © 2013 London Mathematical Society