Related papers: Large Random Simplicial Complexes, II; the fundame…
We investigate the "two-parameter" quantum symmetry groups that we previously constructed with Skalski, with the conclusion that some of these quantum groups, namely those without singletons, are "super-easy" in a suitable sense, that we…
In this paper we study the Linial-Meshulam model of random two-dimensional complexes. We prove that a random 2-complex is homotopically one dimensional, with probability tending to one as n tends to infitnity, assuming that the probability…
We introduce the notion of strictly systolic angled complexes. They generalize Januszkiewickz and \'Swi\k{a}tkowski's $7$-systolic simplicial complexes and also their metric counterparts, which appear as natural analogues to Huang and…
It is proved that fundamental groups of boolean representable simplicial complexes are free and the rank is determined by the number and nature of the connected components of their graph of flats for dimension $\geq 2$. In the case of…
In any dimension at least five we construct examples of closed smooth manifolds with the following properties: 1) they have neither real projective nor flat conformal structures; 2) their fundamental group is a non-elementary Gromov…
We prove that every finitely generated group with recursive aspherical presentation embeds into a group with finite aspherical presentation. This and several known facts about groups and manifolds imply that there exists a 4-dimensional…
We prove that for any group of the cohomological dimension $n$ the $n$th power of the Berstein class of the group is nontrivial. This allows to prove the following Berstein-Svarc theorem for all $n$: Theorem. For a connected complex $X$…
The number of nonisomorphic simplicial complexes with up to $n$ vertices increases super-exponentially with $n$, which makes exhaustive computation of invariants associated with such complexes a daunting task. In this paper we provide a…
We introduce a general framework allowing the systematic study of random manifolds. In order to do so, we will put ourselves in a more general context than usual by allowing the underlying probability space to be non commutative. We…
We prove some triviality results for reduced Whitehead groups and reduced unitary Whitehead groups for division algebras over a henselian discrete valuation field whose residue field has virtual cohomological dimension or separable…
Asymptotic normality is frequently observed in large combinatorial structures, rigorously established for many quantities such as cycles or inversions in random permutations, the number of prime factors of random integers, and various…
We find sharp upper bounds on the order of the automorphism group of a hypersurface in complex projective space in every dimension and degree. In each case, we prove that the hypersurface realizing the upper bound is unique up to…
The spectra of random feature matrices provide essential information on the conditioning of the linear system used in random feature regression problems and are thus connected to the consistency and generalization of random feature models.…
We study the large-scale geometry of graph braid groups $\mathbb{B}_n(\mathsf{\Gamma})$, viewed as the fundamental groups of discrete configuration spaces $UD_n(\mathsf{\Gamma})$, which are special cube complexes in the sense of…
A simplicial complex is $r$-conic if every subcomplex of at most $r$ vertices is contained in the star of a vertex. A $4$-conic complex is simply connected. We prove that an $8$-conic complex is $2$-connected. In general a $(2n+1)$-conic…
In this paper, we study some large scale properties of the mother groups of bounded automata groups. First we give two methods to prove every mother group has infinite asymptotic dimension. Then we study the decomposition complexity of…
The classical random matrix theory is mostly focused on asymptotic spectral properties of random matrices as their dimensions grow to infinity. At the same time many recent applications from convex geometry to functional analysis to…
We introduce the notion of \"uber-contracting element, a strengthening of the notion of strongly contracting element which yields a particularly tractable criterion to show the acylindrical hyperbolicity, and thus a strong form of…
We consider a generalised model of a random simplicial complex, which arises from a random hypergraph. Our model is generated by taking the downward-closure of a non-uniform binomial random hypergraph, in which for each $k$, each set of…
We develop a limit theory for $1$-cochains of complete graphs with coefficients from a finite abelian group. We prove an analogue of the large deviation principle of Chatterjee and Varadhan for random cochains. We use these new tools to…