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We generalize the positive solution of the Frobenius conjecture and refinements thereof by studying the structure of groups that admit a fix-point-free automorphism satisfying an identity. We show, in particular, that for every polynomial…
6-dimensional superconformal field theories are exotic and fascinating. They emerge from compactifications of F-theory on Calabi-Yau elliptic fibrations, which grants them a rich array of dualities with various other formulations of string…
In this article, we prove that if a finitely generated group $G$ is not torsion then a necessary and sufficient condition for every full shift over $G$ has (continuous) cocycle superrigidity is that $G$ has one end. It is a topological…
The concept of gyrogroups, with a weaker algebraic structure without associative law, was introduced under the background of $c$-ball of relativistically admissible velocities with Einstein velocity addition. A topological gyrogroup is just…
Semistability at infinity is an asymptotic property of finitely presented groups that is needed in order to effectively define the fundamental group at infinity for a 1-ended group. It is an open problem whether or not all finitely…
A discrete subset $S$ of a topologically gyrogroup $G$ is called a {\it suitable set} for $G$ if $S\cup \{1\}$ is closed and the subgyrogroup generated by $S$ is dense in $G$, where $1$ is the identity element of $G$. In this paper, we…
In this note we investigate the structure of the space $\Jj$ of smooth almost complex structures on $S^2\times S^2$ that are compatible with some symplectic form. This space has a natural stratification that changes as the cohomology class…
For group presentations with cyclic symmetry, there is a connection between asphericity and the dynamics of the shift automorphism. For the class of groups $G_n(k,l)$ described by the cyclic presentations $\mathcal{P}_n(k,l) =…
Let $G$ be a finitely presented group, and let $H$ be a subgroup of $G$. We prove that if $H$ is acylindrically hyperbolic and existentially closed in $G$, then $G$ is acylindrically hyperbolic. As a corollary, any finitely presented group…
There exist two conjectures for constraint satisfaction problems (CSPs) of reducts of finitely bounded homogeneous structures: the first one states that tractability of the CSP of such a structure is, when the structure is a model-complete…
Let $G$ be a group. \textit{The permutability graph of cyclic subgroups of $G$}, denoted by $\Gamma_c(G)$, is a graph with all the proper cyclic subgroups of $G$ as its vertices and two distinct vertices in $\Gamma_c(G)$ are adjacent if and…
We present a new notion of non-positively curved groups: the collection of discrete countable groups acting (AU-)acylindrically on finite products of $\delta$-hyperbolic spaces with general type factors. Inspired by the classical theory of…
The algebraic dichotomy conjecture for Constraint Satisfaction Problems (CSPs) of reducts of (infinite) finitely bounded homogeneous structures states that such CSPs are polynomial-time tractable if the model-complete core of the template…
We prove the fibred Farrell--Jones Conjecture (FJC) in $A$-, $K$-, and $L$-theory for a large class of suspensions of relatively hyperbolic groups, as well as for all suspensions of one-ended hyperbolic groups. We deduce two applications:…
Recent results show that a constraint satisfaction problem (CSP) defined over rational numbers with their natural ordering has a solution if and only if it has a definable solution. The proof uses advanced results from topology and modern…
We initiate the study of torsion-free algebraically hyperbolic groups; these groups generalise torsion-free hyperbolic groups and are intricately related to groups with no Baumslag--Solitar subgroups. Indeed, for groups of cohomological…
We generalise the definition of a group algebra so that it makes sense for non-locally compact topological groups, in particular, we require that the representation theory of the group algebra is isomorphic (in the sense of Gelfand-Raikov)…
We relate the topology of the Morse boundary of a group to geometric and algorithmic properties of the group. In particular, we show that a group has $\sigma$-compact Morse boundary if and only if it is Morse local-to-global. We also…
If X and Y are orthogonal hyperdefinable sets such that X is simple, then any group G interpretable in (X,Y) has a normal hyperdefinable X-internal subgroup N such that G/N is Y-internal; N is unique up to commensurability. In order to make…
A generalized Baumslag-Solitar group (GBS group) is a finitely generated group $G$ which acts on a tree with all edge and vertex stabilizers infinite cyclic. We show that Out(G) either contains non-abelian free groups or is virtually…