Related papers: On the topological complexity of toral relatively …

We show that the topological complexity of a finitely generated torsion free hyperbolic group $\pi$ with $\cd\pi=n$ equals $2n$.

We prove for non-elementary torsion-free hyperbolic groups $\Gamma$ and all $r\ge 2$ that the higher topological complexity ${\sf{TC}}_r(\Gamma)$ is equal to $r\cdot \mathrm{cd}(\Gamma)$. In particular, hyperbolic groups satisfy the…

We study the higher (or sequential) topological complexity $\mathrm{TC}_s$ of manifolds with abelian fundamental group. We give sufficient conditions for $\mathrm{TC}_s$ to be non-maximal in both the orientable and non-orientable cases. In…

We prove the formula $TC(G\ast H)=\max\{TC(G), TC(H), cd(G\times H)\}$ for the topological complexity of the free product of discrete groups with cohomological dimension >2.

We provide an upper bound on the topological complexity of twisted products. We use it to give an estimate $$TC(X)\le TC(\pi_1(X))+\dim X$$ of the topological complexity of a space in terms of its dimension and the complexity of its…

We note that, for any natural $k$ and every natural $l$ between $k$ and $2k$, there exists a group $\pi$ with $\cat K(\pi,1)=k$ and $\TC(K(\pi,1))=l$. Because of this, we can set up a problem of searching of purely group-theoretical…

We prove that the topological complexity of a finite index subgroup of a hyperbolic group is linear in its index. This follows from a more general result relating the size of the quotient of a free cocompact action of hyperbolic group on a…

We define and develop a homotopy invariant notion for the sequential topological complexity of a map $f:X\to Y,$ denoted $TC_{r}(f)$, that interacts with $TC_{r}(X)$ and $TC_{r}(Y)$ in the same way Jamie Scott's topological complexity map…

We prove the equality $\dTC(\Gamma)=\TC(\Gamma)$ for distributional topological complexity of torsion free hyperbolic and of torsion free nilpotent groups. For the distributional topological complexity of lens spaces we prove the inequality…

The well-known theorem of Eilenberg and Ganea expresses the Lusternik - Schnirelmann category of an aspherical space as the cohomological dimension of its fundamental group. In this paper we study a similar problem of determining…

We prove the formula \begin{equation*} TC(X\vee Y)=\max\{TC(X),TC(Y),cat(X\times Y)\} \end{equation*} for the topological complexity of the wedge $X\vee Y$.

The topological complexity TC(X) is a homotopy invariant which reflects the complexity of the problem of constructing a motion planning algorithm in the space X, viewed as configuration space of a mechanical system. In this paper we…

We define a new version of Topological Complexity (TC) of a space, denoted as $\text{dTC}$, which, we think, fits better for motion planning for some autonomous systems. Like Topological complexity, \text{dTC} is also a homotopy invariant.…

We calculate the higher topological complexity TC$_s$ for the complements of reflection arrangements, in other words for the pure Artin type groups of all finite complex reflection groups. In order to do that we introduce a simple…

We study the complexity of the isomorphism relation for various classes of closed subgroups of the group of permutations of the natural numbers. We use the setting of Borel reducibility between equivalence relations on Polish spaces. For…

A toroidal group is a generalization of a complex torus, and is obtained as the quotient of the complex Euclidean space $\mathbb{C}^n$ by a discrete subgroup. Toroidal groups with finite-dimensional cohomology, called theta toroidal groups,…

Let $\Delta$ be a hyperbolic triangle with a fixed area $\varphi$. We prove that for all but countably many $\varphi$, generic choices of $\Delta$ have the property that the group generated by the $\pi$--rotations about the midpoints of the…

We determine topological complexity of a series of finite spaces which is weakly homotopy equivalent to a circle $S^1$, and give a finite space $X$ satisfying the inequality tc$(X) <$ cat$(X {\times} X)$. This answers two conjectures on…

We prove that a space whose topological complexity equals 1 is homotopy equivalent to some odd-dimensional sphere. We prove a similar result, although not in complete generality, for spaces X whose higher topological complexity TC_n(X) is…

We prove that there is an algorithm to determine whether a tuple of elements in a toral relatively hyperbolic group G is in the automorphic orbit of the other tuple.