arXiv subjects
Publications and source records attributed to Panagiotis Tselekidis.
Inspired by a classical theorem of topological dimension theory, we prove that every geodesic metric space of asymptotic dimension $n$ containing a bi-infinite geodesic can be coarsely separated by a subset $S$ of asymptotic dimension equal to or smaller than $n-1$.\\ We define asymptotic Cantor manifolds, and we prove that every finitely generated group contains such a manifold. We also state some questions related to them.
If $A_Γ$ ($W_Γ$) is the Artin (Coxeter) group with defining graph $Γ$ we denote by $Sim(Γ)$ the number of vertices of the largest clique in $Γ$. We show that $asdimA_Γ\leq Sim(Γ)$, if $Sim(Γ)=2$. We conjecture that the inequality holds for every Artin group. We prove that if for all free of infinity Artin (Coxeter) groups the conjecture holds, then it holds for all Artin (Coxeter) groups. As a corollary, we show that $asdimW_Γ\leq Sim(Γ)$ for all Coxeter groups, which is the best known upper bound for the asymptotic dimension of Coxeter Groups. As a further corollary, we show that the asymptotic dimension of any Artin group of large type with $Sim(Γ)=3$ is exactly two.
Let $W_Γ$ be the Right-Angled Coxeter group with defining graph $Γ$. We show that the asymptotic dimension of $W_Γ$ is smaller than or equal to $dim_{CC}(Γ)$, the clique-connected dimension of the graph. As a corollary we show that $W_Γ$ is virtually free if and only if $dim_{CC}(Γ)=1$.
We prove a new inequality for the asymptotic dimension of HNN-extensions. We deduce that the asymptotic dimension of every finitely generated one relator group is at most two, confirming a conjecture of A.Dranishnikov. As further corollaries we calculate the exact asymptotic dimension of Right-angled Artin groups and we give a new upper bound for the asymptotic dimension of fundamental groups of graphs of groups.