Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userDiscrete Coxeter groups
119
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
Coxeter groups are a special class of groups generated by involutions. They play important roles in the various areas of mathematics. This survey particularly focuses on how one use Coxeter groups to construct interesting examples of discrete subgroups of Lie group.
rate research
Read More
In the 1990s, J.H. Conway published a combinatorial-geometric method for analyzing integer-valued binary quadratic forms (BQFs). Using a visualization he named the topograph, Conway revisited the reduction of BQFs and the solution of quadratic Diophantine equations such as Pells equation. It appears that the crux of his method is the coincidence between the arithmetic group $PGL_2({mathbb Z})$ and the Coxeter group of type $(3,infty)$. There are many arithmetic Coxeter groups, and each may have unforeseen applications to arithmetic. We introduce Conways topograph, and generalizations to other arithmetic Coxeter groups. This includes a study of arithmetic flags and variants of binary quadratic forms.
We prove that certain families of Coxeter groups and inclusions $W_1hookrightarrow W_2hookrightarrow...$ satisfy homological stability, meaning that in each degree the homology $H_ast(BW_n)$ is eventually independent of $n$. This gives a uniform treatment of homological stability for the families of Coxeter groups of type $A_n$, $B_n$ and $D_n$, recovering existing results in the first two cases, and giving a new result in the third. The key step in our proof is to show that a certain simplicial complex with $W_n$-action is highly connected. To do this we show that the barycentric subdivision is an instance of the basic construction, and then use Daviss description of the basic construction as an increasing union of chambers to deduce the required connectivity.
We prove the meridional rank conjecture for twisted links and arborescent links associated to bipartite trees with even weights. These links are substantial generalizations of pretzels and two-bridge links, respectively. Lower bounds on meridional rank are obtained via Coxeter quotients of the groups of link complements. Matching upper bounds on bridge number are found using the Wirtinger numbers of link diagrams, a combinatorial tool developed by the authors.
Given a Coxeter system (W,S), there is an associated CW-complex, Sigma, on which W acts properly and cocompactly. We prove that when the nerve L of (W,S) is a flag triangulation of the 3-sphere, then the reduced $ell^2$-homology of Sigma vanishes in all but the middle dimension.
Let W be an arbitrary Coxeter group. If two elements have expressions that are cyclic shifts of each other (as words), then they are conjugate (as group elements) in W. We say that w is cyclically fully commutative (CFC) if every cyclic shift of any reduced expression for w is fully commutative (i.e., avoids long braid relations). These generalize Coxeter elements in that their reduced expressions can be described combinatorially by acyclic directed graphs, and cyclically shifting corresponds to source-to-sink
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
