Local Tameness and Ends of Groups
We investigate connections between local tameness of a group and a number of its ends.
math.GR↗
arXiv subjects
Rita Gitik
Publications and source records attributed to Rita Gitik.
Optimal Seeding and Self-Reproduction from a Mathematical Point of View
P. Kabamba developed generation theory as a tool for studying self-reproducing systems. We provide an alternative definition of a generation system and give a complete solution to the problem of finding optimal seeds for a finite self-replicating system. We also exhibit examples illustrating a connection between self-replication and fixed-point theory.
cs.AI↗
Closed Form Analytical Model for Airflow around 2-Dimensional Composite Airfoil Via Conformal Mapping
This paper presents a method of computing section lift characteristics for a 2-dimensional airfoil with a second 2-dimensional object at a position at or ahead of the leading edge of the airfoil. Since both objects are 2-dimensional, the analysis yields a closed form solution to calculation of the airflow over the airfoil and second object, using conformal mapping of analytically closed form airflow velocity vector past two circular shaped objects in initial complex plane, using a standard air flow model for each object individually. The combined airflow velocity vector is obtained by linear superposition of the velocity vector for the two objects, computed individually. The lift characteristics are obtained from the circulation around the airfoil and second object which is computed from the combined closed form velocity vector and the geometry along the contour integral for circulation. The illustrative example considered in this paper shows that the second object which is essentially a cylinder whose diameter is approximately 9% of the chord length of the airfoil reduces the section lift coefficient by approximately 6:3% from that of the airfoil alone. 1.
physics.flu-dyn↗
On Growth of Double Cosets in Hyperbolic Groups
Let $H$ be a hyperbolic group, $A$ and $B$ be subgroups of $H$, and $gr(H,A,B)$ be the growth function of the double cosets $AhB, h \in H$. We prove that the behavior of $gr(H,A,B)$ splits into two different cases. If $A$ and $B$ are not quasiconvex, we obtain that every growth function of a finitely presented group can appear as $gr(H,A,B)$. We can even take $A=B$. In contrast, for quasiconvex subgroups A and B of infinite index, $gr(H,A,B)$ is exponential. Moreover, there exists a constant $λ> 0$, such that $gr(H,A,B)(r) >λf_H(r)$ for all big enough $r$, where $f_H(r)$ is the growth function of the group $H$. So, we have a clear dychotomy between the quasiconvex and non-quasiconvex case.
math.GR↗
On The Structure of Dyck Languages
We prove that the closure of the one-sided Dyck language in a free monoid is a two-sided Dyck language.
cs.FL↗
On Products of Closed Subsets in Free Groups
We present examples of closed subsets of a free group such that their product is not closed in the profinite topology. We discuss how to characterize a subset of a free group which is closed in the profinite topology and its product with any finitely generated subgroup of a free group is also closed in the profinite topology.
math.GR↗
On Geodesic Triangles in Hyperbolic Plane
Let M be an orientable hyperbolic surface without boundary and let $γ$ be a closed geodesic in M. We prove that any side of any triangle formed by distinct lifts of $γ$ in H2 is shorter than $γ$.
math.GR↗
Quasi-positivity and recognition of products of conjugacy classes in free groups
Given a group $G$ and a subset $X \subset G$, an element $g \in G$ is called quasi-positive if it is equal to a product of conjugates of elements in the semigroup generated by $X$. This notion is important in the context of braid groups, where it has been shown that the closure of quasi-positive braids coincides with the geometrically defined class of $\mathbb{C}$-transverse links. We describe an algorithm that recognizes whether or not an element of a free group is quasi-positive with respect to a basis. Spherical cancellation diagrams over free groups are used to establish the validity of the algorithm and to determine the worst-case runtime.
math.GR↗
On Local Tameness of Certain Graphs of Groups
Let $G$ be the fundamental group of a finite graph of groups with Noetherian edges and locally tame vertices. We prove that $G$ is locally tame. It follows that if a finitely presented group $H$ has a non-trivial $JSJ$-decomposition over the class of its $VPC(k)$ subgroups for $k=1$ or $k=2$, and all the vertex groups in the decomposition are flexible, then $H$ is locally tame.
math.GR↗
Two Algorithms in Group Theory
We present a new algorithm deciding if the intersection of a quasiconvex subgroup of a negatively curved group with a conjugate is finite. We also give a short proof of decidability of the membership problem for quasiconvex subgroups of finitely generated groups with decidable word problem.
math.GR↗
Conjugate Words and Intersections of Geodesics in $H^2$
We investigate intersections of geodesic lines in $H^2$ and in an associated tree T, proving the following result. Let M be a punctured hyperbolic torus and let $γ$ be a closed geodesic in M. Any edge of any triangle formed by distinct geodesic lines in the preimage of $γ$ in $H^2$ is shorter then $γ$. However, a similar result does not hold in the tree T. Let W be a reduced and cyclically reduced word in $π_1(M) = $. We construct several examples of triangles in T formed by distinct axes in T stabilized by conjugates of W such that an edge in those triangles is longer than L(W). We also prove that if W overlaps two of its conjugates in such a way that the overlaps cover all of W and the overlaps do not intersect, then there exists a decomposition $W = BC^kI; k > 0$, with B a terminal subword of C and I an initial subword of C.
math.GR↗
A Generalization of a Power-Conjugacy Problem in Torsion-Free Negatively Curved Groups
Let H and K be quasiconvex subgroups of a negatively curved torsion-free group G. We give an algorithm which decides whether an element of H is conjugated in G to an element of K.
math.GR↗
On Tame Subgroups of Finitely Presented Groups
We prove that the free product of two finitely presented locally tame groups is locally tame and describe many examples of tame subgroups of finitely presented groups. We also include some open problems related to tame subgroups.
math.GR↗
On Closed Subsets of Free Groups
We give two examples of a finitely generated subgroup of a free group and a subset, closed in the profinite topology of a free group, such that their product is not closed in the profinite topology of a free group.
math.GR↗
On Intersection of Conjugate Subgoups
We give several algorithms addressing computations of intersections of conjugate subgroups.
math.GR↗
Weak Width of Subgroups
We say that the weak width of an infinite subgroup $H$ of $G$ in $G$ is $n$ if there exists a collection of $n$ strongly essentially distinct conjugates $\{ H, g_1^{-1} H g_1,\cdots, g_{n-1}^{-1} H g_{n-1} \}$ of $H$ in $G$ such that the intersection $H \cap g_i^{-1} H g_i$ is infinite for all $1 \leq i \leq n-1$ and $n$ is maximal possible. We prove that a quasiconvex subgroup of a negatively curved group has finite weak width in the ambient group. We also give examples demonstrating that height, width, and weak width are different invariants of a subgroup.
math.GR↗