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We discuss the universal orbifold Euler characteristic and generalized orbifold Euler characteristics corresponding to finitely generated groups $A$ (the $A$-Euler characteristics). We show that the collection of all $A$-Euler characteristics for $A$ of the form $A'\times Z$ ($Z$ is the group of integers) with finite $A'$ determine the universal orbifold Euler characteristic.

The notion of effective topological complexity, introduced by B{\l}aszczyk and Kaluba, deals with using group actions in the configuration space in order to reduce the complexity of the motion planning algorithm. In this article we focus on studying several properties of such notion of topological complexity. We introduce a notion of effective LS-category which mimics the behaviour the usual LS-cat has in the non-effective setting. We use it to investigate the relationship between these effective invariants and the orbit map with respect of the group action, and we give numerous examples. Additionally, we investigate non-vanishing criteria based on a cohomological dimension bound of the saturated diagonal., Comment: 31 pages, 1 figure. Several improvements and corrections of typos, some additional results added. Preliminary version, comments are most welcomed!

We establish that the natural permutation representation of highly transitive groups cannot be realized as the full group of automorphisms of an idempotent finite-dimensional evolution algebra acting on the set of lines spanned by natural elements. For instance, for any sufficiently large integer $n$, and $k\geq 4$, there exists no idempotent evolution algebra $X$ such that $\dim X=n$, and $\operatorname{Aut}(X)$ is isomorphic to a proper $k$-transitive subgroup of $S_n$. However, we demonstrate that for any (not necessarily faithful) permutation representation $\xi\colon G \to S_n$ and any field $\Bbbk$, there exists an idempotent finite-dimensional evolution $\Bbbk$-algebra $X$ such that $\operatorname{Aut}(X)\cong G$ and the induced representation given by the $\operatorname{Aut}(X)$-action on the natural idempotents of $X$ is equivalent to $\xi$., Comment: 14 pages, no figures