Related papers: Stable commutator length on free $\mathbb{Q}$-grou…

In this article, we prove the algebraic counterpart of the topological results $H^1(S^1, \mathbb{Z}) \cong \mathbb{Z}$ and $H^1(S^2, \mathbb{Z}) \cong \{0\}$. We also see that a non-trivial element of the algebraic cohomotopy groups of…

We study uniform stability of discrete groups, Lie groups and Lie algebras in the rank metric, and the connections between uniform stability of these objects. We prove that semisimple Lie algebras are far from being flexibly…

For all $k \ge 2$, we show that there exists a group $G$ and a non-free stably free $\mathbb{Z} G$-module of rank $k$. We use this to show that, for all $k \ge 2$, there exist homotopically distinct finite $2$-complexes with fundamental…

We prove a general homological stability theorem for certain families of groups equipped with product maps, followed by two theorems of a new kind that give information about the last two homology groups outside the stable range. (These…

We compute the homotopy type of the moduli space of flat, unitary connections over aspherical surfaces, after stabilizing with respect to the rank of the underlying bundle. Over the orientable surface M^g, we show that this space has the…

We study the quasi-isometric rigidity of a large family of finitely generated groups that split as graphs of groups with virtually free vertex groups and two-ended edge groups. Let $G$ be a group that is one-ended, hyperbolic relative to…

The existence of stable ACM vector bundles of high rank on algebraic varieties is a challenging problem. In this paper, we study stable Ulrich bundles (that is, stable ACM bundles whose corresponding module has the maximum number of…

A homogeneous quasimorphism $\phi$ on a normal subgroup $N$ of $G$ is said to be $G$-invariant if $\phi(gxg^{-1}) = \phi(x)$ for every $g \in G$ and for every $x \in N$. Invariant quasimorphisms have naturally appeared in symplectic…

Let $q$ be a non-negative integer. We prove that a perfect field $K$ has cohomological dimension at most $q+1$ if, and only if, for any finite extension $L$ of $K$ and for any homogeneous space $Z$ under a smooth linear connected algebraic…

We discuss generalizations of the notion of i) the group of unitary elements of a (real or complex) finite dimensional C*-algebra, ii) gauge transformations and iii) (real) automorphisms, in the framework of compact quantum group theory and…

We study model geometries of finitely generated groups. If a finitely generated group does not contain a non-trivial finite rank free abelian commensurated subgroup, we show any model geometry is dominated by either a symmetric space of…

Motivated by the work of Goswami on quantum isometry groups of noncommutative manifolds we define the quantum symmetry group of a unital C*-algebra A equipped with an orthogonal filtration as the universal object in the category of compact…

Geometric and dynamic properties of embeddings of SL(2,Z) into the Cremona group are studied. Infinitely many non-conjugate embeddings which preserve the type (i.e. which send elliptic, parabolic and hyperbolic elements onto elements of the…

Using a recent result of Bowden, Hensel and Webb, we prove the existence of a homeomorphism with positive stable commutator length in the group of homeomorphisms of the Klein bottle which are isotopic to the identity.

We prove that the existence of a positively defined, invariant Einstein metric $m$ on a connected homogeneous space $G/H$ of a compact Lie group $G$ is the consequence of non-contractibility of some compact set $C=X_{G,H}^{\Sigma}$ (B\"ohm…

We study slope-stable vector bundles and Bridgeland stability conditions on varieties which are a quotient of a smooth projective variety by a finite abelian group $G$ acting freely. We show there is an analytic isomorphism between…

Through highly non-constructive methods, works by Bestvina, Culler, Feighn, Morgan, Paulin, Rips, Shalen, and Thurston show that if a finitely presented group does not split over a virtually solvable subgroup, then the space of its discrete…

Recently, it was shown that Einstein solvmanifolds have maximal symmetry in the sense that their isometry groups contain the isometry groups of any other left-invariant metric on the given Lie group. Such a solvable Lie group is necessarily…

For which groups $G$ is it true that for all fields $k$, every non-monomial element of the group algebra $k\,G$ generates a proper $2$-sided ideal? The only groups for which we know this are the torsion-free abelian groups. We would like to…

We compute the quantum isometry group of the finite noncommutative geometry F describing the internal degrees of freedom in the Standard Model of particle physics. We show that this provides genuine quantum symmetries of the spectral triple…