Random groups are not n-cubulated

TL;DR

This work investigates when random groups act with global fixed points on finite-dimensional $ ext{CAT}(0)$ cube complexes, introducing a robust fixed-point framework $FW_n$ that strengthens the link between Serre's $FA$ and property $(T)$. The authors develop progressing automata and checkpoint-tree machinery to force fixed points by ensuring relators intersect dense Languages of these automata, then transfer these probabilistic intersections to the random-group presentations. They prove that plain-words random groups have $FW_n$ whp for every $n$ and density, and extend the same conclusion to reduced-words models provided the generator set is large enough, yielding the first cubulated hyperbolic groups with $FW_n$ for arbitrarily large $n$. Moreover, they provide a Rips-based construction to obtain $FW_n$ groups and derive an $FW_ ext{∞}$ cubulated example with cohomological dimension $2$, illustrating the depth and versatility of their automata-based approach for generating cubical fixed-point phenomena.

Abstract

A group $G$ has $FW_n$ if every action on a $n$-dimensional $\mathrm{CAT}(0)$ cube complex has a global fixed point. This provides a natural stratification between Serre's $FA$ and Kazhdan's $(T)$. For every $n$, we show that random groups in the plain words density model have $FW_n$ with overwhelming probability. The same result holds for random groups in the reduced words density model assuming there are sufficiently many generators. These are the first examples of cubulated hyperbolic groups with $FW_n$ for $n$ arbitrarily large.

Figures (8)

• Figure 1: $P\to \Sigma$ begins at $\mathfrak{s}$.
• Figure 2: $P\to \Sigma$ begins at a checkpoint vertex $c_1$. The red and blue in (B) denote two possible cases.
• Figure 3: Translates of $x$ by $\mathcal{A}_w(H)$ and $\mathcal{B}_w(H)$.
• Figure 4: Translate of $x$ by $\mathcal{P}^\vert_w(H)$, $\mathcal{P}^\pitchfork_w(H)$, and $\mathcal{P}^\parallel_w(H)$.
• Figure 5: The path $P'$ is labeled $us$. The red subpath is $P'_0$, and the blue edge is $e$. The rooted red path is the lift $P_0$ of $P'_0$. There is a rooted lift of $P$ if and only if there is an edge labeled $s$ at the terminal end of $P_0$.

Theorems & Definitions (69)

• Definition 1.1: $FW_n$, $FW_\infty$, $FW$
• Theorem 1.2
• Theorem 1.3
• Corollary 1.4
• proof
• Theorem 1.5
• Corollary 1.6: $FW_n$ Rips Construction
• Lemma 1.7
• Theorem 2.1
• Lemma 2.2