Tensor product decompositions of II1_1 factors arising from extensions of amalgamated free product groups

In this paper we introduce a new family of icc groups $\Gamma$ which satisfy the following product rigidity phenomenon, discovered in [DHI16] (see also [dSP17]): all tensor product decompositions of the II$_1$ factor $L(\Gamma)$ arise only from the canonical direct product decompositions of the underlying group $\Gamma$. Our groups are assembled from certain HNN-extensions and amalgamated free products and include many remarkable groups studied throughout mathematics such as graph product groups, poly-amalgam groups, Burger-Mozes groups, Higman group, various integral two-dimensional Cremona groups, etc. As a consequence, we obtain several new examples of groups that give rise to prime factors

On graphs and valuations

In the last two decades new techniques emerged to construct valuations on an infinite division ring $D,$ given a normal subgroup $N\subseteq D$ of finite index. These techniques were based on the commuting graph of $D^{\times}/N$ in the case where $D$ is non-commutative, and on the Milnor K-graph on $D^{\times}/N,$ in the case where $D$ is commutative. In this paper we unify these two approaches and consider V-graphs on $D^{\times}/N$ and how they lead to valuations. We furthermore generalize previous results to situations of finitely many valuations.Comment: 40 pages, no figure

On the Diameter of the Commuting Graph of a Full Matrix Ring over a Division Ring

For a division ring D, finite dimensional over its center F, we give a condiction for the connectedness of the commuting graph of a matrix ring over $D$. Furthermore, we prove that if the commuting graph is connected, then its diameter is between four and six

Star reducible Coxeter groups

We define ``star reducible'' Coxeter groups to be those Coxeter groups for which every fully commutative element (in the sense of Stembridge) is equivalent to a product of commuting generators by a sequence of length-decreasing star operations (in the sense of Lusztig). We show that the Kazhdan--Lusztig bases of these groups have a nice projection property to the Temperley--Lieb type quotient, and furthermore that the images of the basis elements $C'_w$ (for fully commutative $w$) in the quotient have structure constants in ${\Bbb Z}^{\geq 0}[v, v^{-1}]$. We also classify the star reducible Coxeter groups and show that they form nine infinite families (types $A_n$, $B_n$, $D_n$, $E_n$, $F_n$, $H_n$, affine $A_{n-1}$ for $n$ odd, affine $C_{n-1}$ for $n$ even, and the case where the Coxeter graph is complete), with two exceptional cases (of ranks 6 and 7). This paper is the sequel to math.QA/0509362.Comment: Approximately 41 pages, AMSTeX, 4 figures. Revised in light of referee comments. To appear in the Glasgow Mathematical Journa

Quantum K-theory on flag manifolds, finite-difference Toda lattices and quantum groups

We conjecture that appropriate K-theoretic Gromov-Witten invariants of complex flag manifolds G/B are governed by finite-difference versions of Toda systems constructed in terms of the Langlands-dual quantized universal enveloping algebras U_q(g'). The conjecture is proved in the case of classical flag manifolds of the series A. The proof is based on a refinement of the famous Atiyah-Hirzebruch argument for rigidity of arithmetical genus applied to hyperquot-scheme compactifications of spaces of rational curves in the flag manifolds.Comment: 25 page

Metabelianisations of finitely presented groups

In this article, I study some classes of finitely presented groups with the aim of finding out whether the maximal metabelian quotients of the members of these classes admit finite presentations. The considered classes include those of soluble groups, of one-relator or knot groups, and of Artin groups.Comment: 34 pages, 1 figure. In this second version, Proposition 7.9 has been added, the analysis of the metabelian tops of irreducible Artin systems of finite type B, carried out in Section 8, has been completed and a number of misprints have been correcte

On distance two in Cayley graphs of Coxeter groups

We consider the Cayley graph ${\rm C}(W,S)$ of a Coxeter system $(W,S)$ and describe all maximal $2$-cliques in this graph, i.e. maximal subsets in the vertex set such that the distance between any two distinct elements is equal to $2$. As an application, we show that every automorphism of the half of Cayley graph is uniquely extendable to an automorphism of the Cayley graph if $|S|\ge 5$

Higher rank graphs, k-subshifts and k-automata

Given a $k$-graph $\Lambda$ we construct a Markov space $M_\Lambda$, and a collection of $k$ pairwise commuting cellular automata on $M_\Lambda$, providing for a factorization of Markov's shift. Iterating these maps we obtain an action of ${\mathbb N}^k$ on $M_\Lambda$ which is then used to form a semidirect product groupoid $M_\Lambda \rtimes {\mathbb N}^k$. This groupoid turns out to be identical to the path groupoid constructed by Kumjian and Pask, and hence its C*-algebra is isomorphic to the higher rank graph C*-algebra of $\Lambda$

A matrix ring with commuting graph of maximal diameter

The commuting graph of a semigroup is the set of non-central elements; the edges are defined as pairs $(u,v)$ satisfying $uv=vu$. We provide an example of a field $F$ and an integer $n$ such that the commuting graph of $\operatorname{Mat}_n(F)$ has maximal possible diameter, equal to six.Comment: 7 pages; a corrected version which is to appear in JCTA. The description of the field was incorrect in the first versio

Infinite groups acting faithfully on the outer automorphism group of a right-angled Artin group

We construct the first known examples of infinite subgroups of the outer automorphism group of Out(A_Gamma), for certain right-angled Artin groups A_Gamma. This is achieved by introducing a new class of graphs, called focused graphs, whose properties allow us to exhibit (infinite) projective linear groups as subgroups of Out(Out(A_Gamma)). This demonstrates a marked departure from the known behavior of Out(Out(A_Gamma)) when A_Gamma is free or free abelian, as in these cases Out(Out(A_Gamma)) has order at most 4. We also disprove a previous conjecture of the second author, producing new examples of finite order members of certain Out(Aut(A_Gamma))