arXiv subjects
Publications and source records attributed to Dessislava H. Kochloukova.
For a group $G$ that is a limit group over Droms RAAGs such that $G$ has trivial center, we show that $Σ^1(G) = \emptyset = Σ^1(G, \mathbb{Q})$. For a group $H$ that is a finitely presented residually Droms RAAG we calculate $Σ^1(H)$ and $Σ^2(H)_{dis}$. In addition, we obtain a necessary condition for $[χ]$ to belong to $Σ^n(H)$.
We develop a version of the Bass-Serre theory for Lie algebras (over a field $k$) via a homological approach. We define the notion of fundamental Lie algebra of a graph of Lie algebras and show that this construction yields Mayer-Vietoris sequences. We extend some well known results in group theory to $\mathbb{N}$-graded Lie algebras: for example, we show that one relator $\mathbb{N}$-graded Lie algebras are iterated HNN extensions with free bases which can be used for cohomology computations and apply the Mayer-Vietoris sequence to give some results about coherence of Lie algebras.
For a finitely generated group $G$ we calculate the Bieri-Neumann-Strebel-Renz invariant $Σ^1(\X(G))$ for the weak commutativity construction $\X(G)$. Identifying $S(\X(G))$ with $S(\X(G) / W(G))$ we show $Σ^2(\X(G),\Z) \subseteq Σ^2(\X(G)/ W(G),\Z)$ and $Σ^2(\X(G)) \subseteq $ $ Σ^2(\X(G)/ W(G))$ that are equalities when $W(G)$ is finitely generated and we explicitly calculate $Σ^2(\X(G)/ W(G),\Z)$ and $ Σ^2(\X(G)/ W(G))$ in terms of the $Σ$-invariants of $G$. We calculate completely the $Σ$-invariants in dimensions 1 and 2 of the group $ν(G)$ and show that if $G$ is finitely generated group with finitely presented commutator subgroup then the non-abelian tensor square $G \otimes G$ is finitely presented.
We study the Bieri-Neumann-Strebel-Renz invariants and we prove the following criterion: for groups $H$ and $K$ of type $FP_n$ such that $[H,H] \subseteq K \subseteq H$ and a character $χ: K \to \mathbb{R}$ with $χ([H,H]) = 0$ we have $[χ] \in Σ^n(K, \mathbb{Z})$ if and only if $[μ] \in Σ^n(H, \mathbb{Z})$ for every character $μ: H \to \mathbb{R}$ that extends $χ$. The same holds for the homotopical invariants $Σ^n(-)$ when $K$ and $H$ are groups of type $F_n$. We use these criteria to complete the description of the $Σ$-invariants of the Bieri-Stallings groups $G_m$ and more generally to describe the $Σ$-invariants of the Bestvina-Brady groups. We also show that the "only if" direction of such criterion holds if we assume only that $K$ is a subnormal subgroup of $H$, where both groups are of type $FP_n$. We apply this last result to wreath products.
We calculate the Bieri-Neumann-Strebel-Renz invariant $Σ^1(G)$ for even Artin groups $G$ with underlying graph $Γ$ such that if there is a closed reduced path in $Γ$ with all labels bigger than 2 then the length of such path is always odd. We show that $Σ^1(G)^c$ is a rationally defined spherical polyhedron.
We define and study a pro-$p$ version of Sidki's weak commutativity construction. This is the pro-$p$ group $\mathfrak{X}_p(G)$ generated by two copies $G$ and $G^ψ$ of a pro-$p$ group, subject to the defining relators $[g,g^ψ]$ for all $g \in G$. We show for instance that if $G$ is finitely presented or analytic pro-$p$, then $\mathfrak{X}_p(G)$ has the same property. Furthermore we study properties of the non-abelian tensor product and the pro-$p$ version of Rocco's construction $ν(H)$. We also study finiteness properties of subdirect products of pro-$p$ groups. In particular we prove a pro-$p$ version of the $(n-1)-n-(n+1)$ Theorem.
We introduce the notion of virtual endomorphisms of Lie algebras and use it as an approach for constructing self-similarity of Lie algebras. This is done in particular for a class of metabelian Lie algebras having homological type F Pn, which are variants of lamp-lighter groups. We establish several criteria when the existence of virtual endomorphism implies a self-similar Lie structure. Furthermore, we prove that the classical Lie algebra sln(k), where char(k) does not divide n affords non-trivial faithful self-similarity.
We show Lie algebra versions of some results on homological finiteness properties of subdirect products of groups, including a version of the 1-2-3 Theorem.
We construct new classes of self-similar groups : S-aritmetic groups, affine groups and metabelian groups. Most of the soluble ones are finitely presented and of type FP_{n} for appropriate n.
We study the homological finiteness property FPn of fibre products
We study the asymptotic growth of homology groups and the cellular volume of classifying spaces as one passes to normal subgroups $G_n<G$ of increasing finite index in a fixed finitely generated group $G$, assuming $\bigcap_n G_n =1$. We focus in particular on finitely presented residually free groups, calculating their $\ell_2$ betti numbers, rank gradient and asymptotic deficiency. If $G$ is a limit group and $K$ is any field, then for all $j\ge 1$ the limit of $\dim H_j(G_n,K)/[G,G_n]$ as $n\to\infty$ exists and is zero except for $j=1$, where it equals $-χ(G)$. We prove a homotopical version of this theorem in which the dimension of $\dim H_j(G_n,K)$ is replaced by the minimal number of $j$-cells in a $K(G_n,1)$; this includes a calculation of the rank gradient and the asymptotic deficiency of $G$. Both the homological and homotopical versions are special cases of general results about the fundamental groups of graphs of {\em{slow}} groups. We prove that if a residually free group $G$ is of type $\rm{FP}_m$ but not of type $\rm{FP}_{\infty}$, then there exists an exhausting filtration by normal subgroups of finite index $G_n$ so that $\lim_n \dim H_j (G_n, K) / [G : G_n] = 0 \hbox{for} j \leq m$. If $G$ is of type $\rm{FP}_{\infty}$, then the limit exists in all dimensions and we calculate it.
We show that for every finitely presented pro-$p$ nilpotent-by-abelian-by-finite group $G$ there is an upper bound on $\dim_{\mathbb{Q}_p} (H_1(M, \mathbb{Z}_p) \otimes_{\mathbb{Z}_p} \mathbb{Q}_p )$, as $M$ runs through all pro-$p$ subgroups of finite index in $G$.
We show that if a group G is finitely presented and nilpotent-by-abelian-by-finite, then there is an upper bound on the first betti number of M as M runs through all subgroups of finite index in G.
We develop new invariants similar to the Bieri-Strebel-Neumann-Renz invariants but in the category of Bredon modules (with respect to the class of the finite subgroups of G). We prove that for virtually soluble groups of type FP_{\infty} and finite extension of the Thompson group F the new invariants coincide with the classical ones.
We begin a study of a pro-$p$ analogue of limit groups via extensions of centralizers and call $\mathcal{L}$ this new class of pro-$p$ groups. We show that the pro-$p$ groups of $\mathcal{L}$ have finite cohomological dimension, type $FP_{\infty}$ and non-positive Euler characteristic. Among the group theoretic properties it is proved that they are free-by-(torsion-free poly -procyclic) and if non-abelian do not have a finitely generated non-trivial normal subgroup of infinite index. Furthermore it is shown that every 2 generated pro-$p$ group in the class $\mathcal{L}$ is either free pro-$p$ or abelian.
We show that Brin's generalisations $2V$ and $3V$ of the Thompson-Higman group $V$ are of type $FP_\infty$. Our methods also give a new proof that both groups are finitely presented.