Abstract twisted Brin--Thompson groups

Abstract

Given a group $G$ acting faithfully on a set $S$, one gets a simple group denoted $SV_G$, called a twisted Brin--Thompson group. In this paper we drop the faithfulness assumption, and get an abstract version of a twisted Brin--Thompson group $SV_G$. While the resulting group is not simple, since $SV_G$ surjects onto $SV_{G/\ker(G \curvearrowright S)}$, we prove that every proper normal subgroup of $SV_G$ lies in the kernel of this surjection, so $SV_G$ is ``relatively simple''. The advantage is that now we can prove that every finitely presented simple group embeds in a finitely presented abstract twisted Brin--Thompson group intersecting this kernel trivially. In particular, if the Boone--Higman conjecture is true, then so is a related conjectural characterization of groups with solvable word problem, arising purely in the world of twisted Brin--Thompson groups. We also prove a variety of additional results about abstract twisted Brin--Thompson groups, some of which are new even in the faithful case: they are all uniformly perfect, have property NL and property FW$_\infty$, are boundedly acyclic and $\ell^2$-invisible, and are $C^*$-simple as soon as they have trivial amenable radical. Along the way we formulate a new general criterion for $\ell^2$-invisibility that is interesting in its own right.

Figures (6)

• Figure 1: The multicolored tree $(x_g\oplus 1 \oplus x_r\oplus 1)\circ (x_b\oplus x_b)\circ x_r \colon C^S \to C^S(6)$, where $S=\{r,b,g\}$, with the simple split $x_r$ colored red, $x_b$ blue, and $x_g$ green. It defines an arboreal partition whose blocks $B(\psi_1), \ldots, B(\psi_6)$, ordered bottom to top, satisfy e.g., $\psi_4(r) = 10, \psi_4(b) = 0, \psi_4(g) = \varnothing$.
• Figure 2: The cross relation $(x_b \oplus x_b)\circ x_r = p_{(2~3)}\circ (x_r \oplus x_r)\circ x_b$, with $x_r$ colored red and $x_b$ colored blue, and $p_{(2~3)} \in \mathcal{S}(4)$.
• Figure 3: The relations $x_r\circ g = (g\oplus g)\circ x_b$ and $(g_1\oplus g_2)\circ p_{(1~2)} = p_{(1~2)}\circ (g_2\oplus g_1)$, with $x_r$ colored red, $x_b$ colored blue, and $g$ such that $g.b = r$.
• Figure 4: Two elements of $SV_G$, for $S=\{r,b,g\}$ (represented by red, blue, and green), $g_1,\dots,g_4\in G$, and $k_1,\dots,k_4\in \ker(G\curvearrowright S)$. The element on the right is in $SK_G$ since the two multicolored trees are the same and the labels from $G$ lie in $\ker(G\curvearrowright S)$.
• Figure 5: With $S=\{r,b\}$, the deferment $D_{B(\psi)}(g)$ for $\psi(r)=1$ and $\psi(b)=0$.

Theorems & Definitions (127)

• Definition 1.1: Normal pair
• Definition 1.2: Relatively simple
• Theorem A: Theorem \ref{['thrm:rel_simple']}
• Definition 1.3
• Theorem B: Theorem \ref{['thrm:fin_props']}
• Conjecture 1.4: Boone--Higman (BH)
• Conjecture 1.5: Permutational Boone--Higman (PBH)
• Definition 1.6: Embeds, sharply embeds
• Conjecture 1.7: Relative Boone--Higman (relBH)
• Conjecture 1.8: Relative permutational Boone--Higman (relPBH)