Peano-Gosper curves and the local isomorphism property

We consider unbounded curves without endpoints. Isomorphism is equivalence up to translation. Self-avoiding plane-filling curves cannot be periodic, but they can satisfy the local isomorphism property: We obtain a set $\Omega$ of coverings of the plane by sets of disjoint self-avoiding nonoriented curves, generalizing the Peano-Gosper curves, such that: 1) each $C \in \Omega$ satisfies the local isomorphism property; any set of curves locally isomorphic to $C$ belongs to $\Omega$; 2) $\Omega$ is the union of $2^{\omega }$ equivalence classes for the relation "$C$ locally isomorphic to $D$"; each of them contains $2^{\omega }$ (resp. $2^{\omega }$, $4$, $0$) isomorphism classes of coverings by $1$ (resp. $2$, $3$, $\geq 4$) curves. Each $C \in \Omega$ gives exactly $2$ coverings by sets of oriented curves which satisfy the local isomorphism property. They have opposite orientations.Comment: 15 pages, 5 figure

Self-avoiding and plane-filling properties for terdragons and other triangular folding curves

We consider $n$-folding triangular curves, or $n$-folding t-curves, obtained by folding $n$ times a strip of paper in $3$, each time possibly left then right or right then left, and unfolding it with $\pi /3$ angles. An example is the well known terdragon curve. They are self-avoiding like $n$-folding curves obtained by folding $n$ times a strip of paper in two, each time possibly left or right, and unfolding it with $\pi /2$ angles. We also consider complete folding t-curves, which are the curves without endpoint obtained as inductive limits of $n$-folding t-curves. We show that each of them can be extended into a unique covering of the plane by disjoint such curves, and this covering satisfies the local isomorphism property introduced to investigate aperiodic tiling systems. Two coverings are locally isomorphic if and only if they are associated to the same sequence of foldings. Each class of locally isomorphic coverings contains exactly $2^{\omega }$ (resp. $2^{\omega }$, $2$ or $5$, $0$) isomorphism classes of coverings by $1$ (resp. $2$, $3$, $\geq 4$) curves. These properties are partly similar to those of complete folding curves.Comment: 15 pages, 3 figure

Decorated hypertrees

C. Jensen, J. McCammond and J. Meier have used weighted hypertrees to compute the Euler characteristic of a subgroup of the automorphism group of a free product. Weighted hypertrees also appear in the study of the homology of the hypertree poset. We link them to decorated hypertrees after a general study on decorated hypertrees, which we enumerate using box trees.---C. Jensen, J. McCammond et J. Meier ont utilis\'e des hyperarbres pond\'er\'es pour calculer la caract\'eristique d'Euler d'un sous-groupe du groupe des automorphismes d'un produit libre. Un autre type d'hyperarbres pond\'er\'es appara\^it aussi dans l'\'etude de l'homologie du poset des hyperarbres. Nous \'etudions les hyperarbres d\'ecor\'es puis les comptons \`a l'aide de la notion d'arbre en bo\^ite avant de les relier aux hyperarbres pond\'er\'es.Comment: nombre de pages : 3

Hypertree posets and hooked partitions

We adapt here the computation of characters on incidence Hopf algebras introduced by W. Schmitt in the 1990s to a family mixing bounded and unbounded posets. We then apply our results to the family of hypertree posets and partition posets. As a consequence, we obtain some enumerative formulas and a new proof for the computation of the Moebius numbers of the hypertree posets. Moreover, we compute the coproduct of the incidence Hopf algebra and recover a known formula for the number of hypertrees with fixed valency set and edge sizes set.Comment: 18 page

Semi-pointed partition posets and Species

We define semi-pointed partition posets, which are a generalisation of partition posets and show that they are Cohen-Macaulay. We then use multichains to compute the dimension and the character for the action of the symmetric groups on their homology. We finally study the associated incidence Hopf algebra, which is similar to the Fa{\`a} di Bruno Hopf algebra.Comment: 27 page

Direct products and elementary equivalence of polycyclic-by-finite groups

We give an algebraic characterization of elementary equivalence for polycyclic-by-finite groups. Using this characterization, we investigate the relations between their elementary equivalence and the elementary equivalence of the factors in their decompositions in direct products of indecomposable groups. In particular we prove that the elementary equivalence of two such groups G,H is equivalent to each of the following properties: 1)Gx...xG (k times G) and Hx...xG (k times H) are elementarily equivalent for a strictly positive integer k; 2)AxG and AxH are elementarily equivalent for two elementarily equivalent polycyclic-by-finite groups A,B. It is not presently known if 1) implies elementary equivalence for any groups G,H.Comment: 15 pages. Minor changes in pages 1 to 3, following the remarks of a referee. The paper is presently publishe