Discrete Morse theory and graph braid groups

Separation in the BNSR-invariants of the pure braid groups

We inspect the BNSR-invariants $Σ^m(P_n)$ of the pure braid groups $P_n$, using Morse theory. The BNS-invariants $Σ^1(P_n)$ were previously computed by Koban, McCammond and Meier. We prove that for any $3\le m\le n$, the inclusion $Σ^{m-2}(P_n)\subseteq Σ^{m-3}(P_n)$ is proper, but $Σ^\infty(P_n)=Σ^{n-2}(P_n)$. We write down explicit character classes in each relevant $Σ^{m-3}(P_n)\setminus Σ^{m-2}(P_n)$. In particular we get examples of normal subgroups $N\le P_n$ with $P_n/N\cong\mathbb{Z}$ such that $N$ is of type $F_{m-3}$ but not $F_{m-2}$, for all $3\le m\le n$.

Morse Theory for Complexes of Groups

We construct an equivariant version of discrete Morse theory for simplicial complexes endowed with group actions. The key ingredient is a 2-categorical criterion for making acyclic partial matchings on the quotient space compatible with an overlaid complex of groups. We use the discrete flow category of any such compatible matching to build the corresponding Morse complex of groups. Our main result establishes that the development of the Morse complex of groups recovers the original simplicial complex up to equivariant homotopy equivalence.

Morse actions of discrete groups on symmetric space

We study the geometry and dynamics of discrete infinite covolume subgroups of higher rank semisimple Lie groups. We introduce and prove the equivalence of several conditions, capturing "rank one behavior'' of discrete subgroups of higher rank Lie groups. They are direct generalizations of rank one equivalents to convex cocompactness. We also prove that our notions are equivalent to the notion of Anosov subgroup, for which we provide a closely related, but simplified and more accessible reformulation, avoiding the geodesic flow of the group. We show moreover that the Anosov condition can be rel…

Equivariant Morse theory on Vietoris-Rips complexes & universal spaces for proper actions

We formalize an equivariant version of Bestvina-Brady discrete Morse theory, and apply it to Vietoris-Rips complexes in order to exhibit finite universal spaces for proper actions for all asymptotically CAT(0) groups.

离散群在对称空间上的莫尔斯作用：局部到整体原理

Morse actions of discrete groups on symmetric spaces: Local-to-global principle

本文建立了莫尔斯拟测地线、映射和作用的局部到整体原理，并据此证明莫尔斯作用的算法可识别性，同时用纯几何方法构造了高阶半单李群的莫尔斯肖特基子群。

$p$-子群复形的轨道空间的可缩性：基于Brown-Forman离散Morse理论的证明

Contractibility of the orbit space of the $p$-subgroup complex via Brown-Forman discrete Morse theory

通过Brown-Forman离散Morse理论，简洁证明了有限群$p$-子群复形轨道空间的可缩性，该结果原由Webb猜想、Symonds证明。

Bestvina-Brady discrete Morse theory and Vietoris-Rips complexes

We inspect Vietoris-Rips complexes $VR_t(X)$ of certain metric spaces $X$ using a new generalization of Bestvina-Brady discrete Morse theory. Our main result is a pair of metric criteria on $X$, called the Morse Criterion and Link Criterion, that allow us to deduce information about the homotopy types of certain $VR_t(X)$. One application is to topological data analysis, specifically persistence of homotopy type for certain Vietoris-Rips complexes. For example we recover some results of Adamaszek-Adams and Hausmann regarding homotopy types of $VR_t(S^n)$. Another application is to geometric gr…

A note on degenerations of Morse actions

We study Morse representations of discrete subgroups in higher rank semi-simple Lie groups defined by M. Kapovich, B. Leeb and J. Porti. We show that, if a sequence of Morse representations $ρ_n : Γ\rightarrow G$ is (strongly) unbounded in the character variety, the group must have a very particular structure.

A Morse Lemma for quasigeodesics in symmetric spaces and euclidean buildings

We prove a Morse Lemma for coarsely regular quasigeodesics in nonpositively curved symmetric spaces and euclidean buildings X. The main application is a simpler coarse geometric characterization of Morse subgroups of the isometry groups Isom(X) as undistorted subgroups which are coarsely uniformly regular. We show furthermore that they must be word hyperbolic. We introduced this class of discrete subgroups in our earlier paper, in the context of symmetric spaces, where various equivalent geometric and dynamical characterizations of word hyperbolic Morse subgroups were established, including th…