Thin buildings

A quasi-isometry invariant and thickness bounds for right-angled Coxeter groups

We introduce a new quasi-isometry invariant of 2-dimensional right-angled Coxeter groups, the hypergraph index, that partitions these groups into infinitely many quasi-isometry classes, each containing infinitely many groups. Furthermore, the hypergraph index of any right-angled Coxeter group can be directly computed from the group's defining graph. The hypergraph index yields an upper bound for a right-angled Coxeter group's order of thickness, order of algebraic thickness and divergence function. Finally, given an integer n>1, we give examples of right-angled Coxeter groups which are thick o…

Characterizing divergence and thickness in right-angled Coxeter groups

We completely classify the possible divergence functions for right-angled Coxeter groups (RACGs). In particular, we show that the divergence of any such group is either polynomial, exponential or infinite. We prove that a RACG is strongly thick of order k if and only if its divergence function is a polynomial of degree k+1. Moreover, we show that the exact divergence function of a RACG can easily be computed from its defining graph by an invariant we call the hypergraph index.

偏移量、厚度和超图指数用于一般Coxeter群

Divergence, thickness and hypergraph index for general Coxeter groups

研究一般Coxeter群的偏移量和厚度，定义可计算的超图指数，证明有限超图指数意味着厚度和偏移量的多项式界，提出拓扑学与超图指数的关系。

Thickness, relative hyperbolicity, and randomness in Coxeter groups

For right-angled Coxeter groups $W_Γ$, we obtain a condition on $Γ$ that is necessary and sufficient to ensure that $W_Γ$ is thick and thus not relatively hyperbolic. We show that Coxeter groups which are not thick all admit canonical minimal relatively hyperbolic structures; further, we show that in such a structure, the peripheral subgroups are both parabolic (in the Coxeter group-theoretic sense) and strongly algebraically thick. We exhibit a polynomial-time algorithm that decides whether a right-angled Coxeter group is thick or relatively hyperbolic. We analyze random graphs in the Erdós-R…

Thin right-angled Coxeter groups in some uniform arithmetic lattices

Using a variant of an unpublished argument due to Agol, we show that an irreducible right-angled Coxeter group on ${n \geq 3}$ vertices embeds as a thin subgroup of a uniform arithmetic lattice in an indefinite orthogonal group $\mathrm{O}(p, q)$ for some $p, q \geq 1$ satisfying $p+q=n$.

Weighted $L^2$-cohomology of Coxeter groups

Given a Coxeter system $(W,S)$ and a positive real multiparameter $\bq$, we study the "weighted $L^2$-cohomology groups," of a certain simplicial complex $Σ$ associated to $(W,S)$. These cohomology groups are Hilbert spaces, as well as modules over the Hecke algebra associated to $(W,S)$ and the multiparameter $q$. They have a "von Neumann dimension" with respect to the associated "Hecke - von Neumann algebra," $N_q$. The dimension of the $i^th$ cohomology group is denoted $b^i_q(Σ)$. It is a nonnegative real number which varies continuously with $q$. When $q$ is integral, the $b^i_q(Σ)$ are t…