arXiv research

Study simplicial volume in fiber bundles with connected groups.

Formula connects foliated simplicial volume with group cost.

The study shows simplicial volume finiteness for certain manifolds with amenable fundamental groups.

The study examines conditions for minimal volume entropy of simplicial complexes.

The simplicial complexity is an invariant for finitely presentable groups that was recently introduced by Babenko, Balacheff and Bulteau to study systolic area. The simplicial complexity $κ(G)$ was proved to be a good approximation of the systolic area $σ(G)$ for large values of $κ(G)$. In this paper we compute the sim…

New techniques prove bounded acyclicity results for semi-simplicial sets.

Study simplicial volume for fixed fundamental groups, finding gaps.

Paper studies spaces of flattenings of simplicial spheres and their homotopy type.

We consider the relation between simplicial volume and two of its variants: the stable integral simplicial volume and the integral foliated simplicial volume. The definition of the latter depends on a choice of a measure preserving action of the fundamental group on a probability space. We show that integral foliated s…

Paper proves Gromov's conjecture on manifolds with certain group properties.

Integral foliated simplicial volume is zero for certain amenable covers.

Minimal volume entropy vanishes or is positive under certain fiber growth conditions.

We introduce the stable presentation length of a finitely presented group. The stable presentation length of the fundamental group of a 3-manifold can be considered as an analogue of the simplicial volume. We show that the stable presentation length have some additive properties like the simplicial volume, and the simp…

This paper explores Brunnian twin groups and their properties.

Integral foliated simplicial volume is a version of simplicial volume combining the rigidity of integral coefficients with the flexibility of measure spaces. In this article, using the language of measure equivalence of groups we prove a proportionality principle for integral foliated simplicial volume for aspherical m…

We discuss nonabelian bundle gerbes and their differential geometry using simplicial methods. Associated to any crossed module there is a simplicial group NC, the nerve of the 1-category defined by the crossed module and its geometric realization |NC|. Equivalence classes of principal bundles with structure group |NC| …

We establish the proportionality principle between the Riemannian volume and locally finite simplicial volume for Q-rank 1 locally symmetric spaces covered by products of hyperbolic spaces, giving the first examples for manifolds whose cusp groups are not necessarily amenable. Also, we give a simple direct proof of the…

Study simplicial volume of manifolds from reflection group trick.

New results on relative simplicial volume using bounded acyclicity.

Study shows complete affine manifolds have zero simplicial volume.

We construct examples of finitely generated groups L that have non-trivial actions on $\mathbb{R}$-trees but which cannot act, without fixing a vertex, on any simplicial tree. Moreover, any finitely presented group mapping onto L does have a fixed point-free action on some simplicial tree.

In this paper, we investigate a relation between finite graphs, simplicial flag complexes and right-angled Coxeter groups, and we provide a class of reconstructible finite graphs. We show that if $Γ$ is a finite graph which is the 1-skeleton of some simplicial flag complex $L$ which is a homology manifold of dimension …

New constructions in group homology allow us to manufacture high-dimensional manifolds with controlled simplicial volume. We prove that for every dimension bigger than 3 the set of simplicial volumes of orientable closed connected manifolds is dense in $\mathbb{R}_{\geq 0}$. In dimension 4 we prove that every non-negat…

We show that integral foliated simplicial volume of closed manifolds gives an upper bound for the cost of the corresponding fundamental groups.

We observe that stable integral simplicial volume of closed manifolds gives an upper bound for the rank gradient of the corresponding fundamental groups.

Link condition for simplicial complexes to be CUB spaces.

Let $M$ be the interior of a connected, oriented, compact manifold $V$ of dimension at least 2. If each path component of $\partial V$ has amenable fundamental group, then we prove that the simplicial volume of $M$ is equal to the relative simplicial volume of $V$ and also to the geometric (Lipschitz) simplicial volume…

New length functions on mapping class groups linked to simplicial volumes of mapping tori.

The paper extends Chern-Weil theory to simplicial principal bundles.

Twenty years ago Gromov asked about how large is the set of isomorphism classes of groups whose systolic area is bounded from above. This article introduces a new combinatorial invariant for finitely presentable groups called {\it simplicial complexity} that allows to obtain a quite satisfactory answer to his question.…

Lie's third theorem proven for Lie ∞-algebras.

This paper contains examples of closed aspherical manifolds obtained as a by-product of recent work by the author [arXiv:math.GR/0509490] on the relative strict hyperbolization of polyhedra. The following is proved. (I) Any closed aspherical triangulated n-manifold M^n with hyperbolic fundamental group is a retract of …

Research explores hyperbolic space groups and their fundamental domains.

Study on simplicial volume and Euler characteristic of aspherical manifolds.

Study minimal volume entropy for free-by-cyclic groups and 2D right-angled Artin groups.

Let $G$ be a higher-rank semisimple Lie group over a nonarchimedean local field, for example $G={\rm PGL}(n,Q_P)$. To any lattice $L$ in $G$ there is an associated simplicial complex $B_L$, given by the quotient by $L$ of the Bruhat-Tits building associated to $G$. In this paper prove that the simplicial structure $B_L…

We prove that every finite connected simplicial complex has the homology of the classifying space for some $\mathrm{CAT}(0)$ cubical duality group. More specifically, for any finite simplicial complex $X$, we construct a locally $\mathrm{CAT}(0)$ cubical complex $T_{X}$ and an acyclic map $t_{X} : T_{X} \to X$ such tha…

Constructs a universal Chern-Weil map for infinite dimensional Lie groups.

We introduce a new homology theory of quandles, called simplicial quandle homology, which is quite different from quandle homology developed by Carter et al. We construct a homomorphism from a quandle homology group to a simplicial quandle homology group. As an application, we obtain a method for computing the complex …

We prove that each injective simplicial map from the arc complex of a compact, connected, nonorientable surface with nonempty boundary to itself is induced by a homeomorphism of the surface. We also prove that the automorphism group of the arc complex is isomorphic to the quotient of the mapping class group of the surf…

In this note on coarse geometry we revisit coarse homotopy. We prove that coarse homotopy indeed is an equivalence relation, and this in the most general context of abstract coarse structures. We introduce (in a geometric way) coarse homotopy groups. The main result is that the coarse homotopy groups of cone of a compa…

Study essentiality and simplicial volume of manifolds fibered over spheres.

New geometric spine for Artin groups defined by cube complexes.

We describe the moduli space of extensions in the model category of simplicial presheaves. This article can be seen as a generalization of Blomgren-Chacholski results in the case of simplicial sets. Our description of the moduli space of extensions treat the equivariant and the nonequivariant case in the same setting. …

Given an n-term L-infinity algebra L, we construct a Kan simplicial manifold which we think of as the 'Lie n-group' integrating L. This extends work of Getzler math.AT/0404003 . In the case of an ordinary Lie algebra, our construction gives the simplicial classifying space of the corresponding simply connect Lie group.…

Let g be a simplicial Lie algebra with Moore complex Ng of length k. Let G be the simplicial Lie group integrating g, which is simply connected in each simplicial level. We use the 1-jet of the classifying space of G to construct, starting from g, a Lie k-algebra L. The so constructed Lie k-algebra L is actually a diff…

The procedure "Lie group --> Lie algebra" has a generalization "simplicial manifold --> L_infinity algebra", or yet better, "presheaf on the category of surjective submersions --> L_infinity algebra". We describe this generalization, together with its higher-order extensions.

In the spirit of topological entropy we introduce new complexity functions for general dynamical systems (namely groups and semigroups acting on closed manifolds) but with an emphasis on the dynamics induced on simplicial complexes. For expansive systems remarkable properties are observed. Known examples are revisited …