Motivated by the work of Leininger on hyperbolic equivalence of homotopy classes of closed curves on surfaces, we investigate a similar phenomenon for free groups. Namely, we study the situation when two elements g, h in a free group F have the property that for every free isometric action of F on an R-tree X the translation lengths of g and h on X are equal.

In a visualization system, one of the key issues is to optimize performance and visual fidelity. This is especially critical for large virtual environments where the models do not fit into the memory. In this paper, we present a novel structure called HDoV-tree that can be tuned to provide excellent visual fidelity and performance based on the degree of visibility of objects. HDoV-tree also exp...

We prove that if T is an R-tree with a minimal free isometric action of FN , then the Out(FN )-stabilizer of the projective class [T ] is virtually cyclic. For the special case where T = T+(φ) is the forward limit tree of an atoroidal iwip element φ ∈ Out(FN ) this is a consequence of the results of Bestvina, Feighn and Handel [6], via very different methods. We also derive a new proof of the T...

of D is an R-tree. (ii) There exists a tree (V, E ) whose vertex set V contains X, and an edge weighting C : E -i R that assigns a positive length C(e) to each edge e in E , such that D is the restriction of X to the shortest-path metric induced on V. (iii) There exists a map w : S(X) -i R>o from the set S(X) of all bi-partitions or splits of X into the set R ~ o of non-negative real numbers su...

‎Unmixed bipartite graphs have been characterized by Ravindra and‎ ‎Villarreal independently‎. ‎Our aim in this paper is to‎ ‎characterize unmixed $r$-partite graphs under a certain condition‎, ‎which is a generalization of Villarreal's theorem on bipartite‎ ‎graphs‎. ‎Also, we give some examples and counterexamples in relevance to this subject‎.