A theorem of Quillen says that if $G$ is a finite Chevalley group over characteristic $p$, then the poset $\mathcal{A}_p(G)$ of nontrivial elementary abelian subgroups of $G$ is homotopy equivalent (I will not distinguish a poset from its order complex much) to the associated Tits building $\Delta(G)$.

I am interested in finding out how far this result can be generalized for a finite $G$ while we can still reasonably talk about an associated building $\Delta(G)$. I shall make this more precise.

Let me start with stuff that does not require any "B,N type" input. Quillen, in the same 1978 paper, observed that for any finite group $G$, if we write $\mathcal{S}_p(G)$ for the poset of all nontrivial $p$-subgroups, then the inclusion $\mathcal{A}_p(G) \hookrightarrow \mathcal{S}_p(G)$ is a homotopy equivalence. This can be seen by observing that the "down-set" $\mathcal{S}_p(G)_{<Q}$ is contractible if $Q$ is outside $\mathcal{A}_p(G)$, and alluding to Quillen's Theorem A.

Applying the same idea dually, Bouc proved that writing $$\mathcal{B}_p(G) := \{Q \in \mathcal{S}_p(G) : Q = O_p(N_G(Q))\}$$

(here $O_p$ denotes the largest normal $p$-subgroup), the "up-set" $\mathcal{S}_p(G)_{>Q}$ is contractible when $Q \notin \mathcal{B}_p(G)$. Therefore the inclusion $\mathcal{B}_p(G) \hookrightarrow \mathcal{S}_p(G)$ is also a homotopy equivalence. Thus we can work with $\mathcal{B}_p(G)$ instead of $\mathcal{A}_p(G)$. Elements of $\mathcal{B}_p(G)$ are often called $p$-radical subgroups in the finite group theory literature. The relationship of $\mathcal{B}_p(G)$ with the building (when there is one) is much more direct than that of $\mathcal{A}_p(G)$, see below.

I don't know which notions about buildings are "standard" and which are not, so I will try to set things up about them carefully without ambiguity. I will start from a general situation in which we can talk about a building and then consider restrictions to be able to compare the building with the intrinsically defined poset $\mathcal{B}_p(G)$.

A general situation to get a building is the following, quoted from Abramenko-Brown's book with some abuse of notation:

Definition 1. Suppose we are given a group $G$, a subgroup $B$, a Coxeter system $(W,S)$, and a bijection $C \colon W \rightarrow B \backslash G/B$ satisfying the following condition:

(B) For all $s \in S$ and $w \in W$, $C(sw) \subseteq C(s)C(w) \subseteq C(sw) \cup C(w)$. If we have $l(sw) = l(w) + 1$, then $C(s)C(w) = C(sw)$.

Then the bijection is said to provide a Bruhat decomposition of type $(W,S)$ for $G$.

In the presence of a Bruhat decomposition, defining $W_J := \langle J \rangle$ and $P_J := \bigcup_{w \in W_J} C(w)$ for each $J \subseteq S$, the $P_J$ turns out to actually be a subgroup and is called a standard parabolic. Moreover, their left cosets $$\Delta(G) := \{gP_J: g \in G,\, J \subsetneq S \} \, ,$$ ordered by reverse inclusion form a building (in the AB book's sense at least).

Definition 2. We say that a pair of subgroups $B$ and $N$ of a group $G$ is a BN-pair if $B$ and $G$ generate $G$, the intersection $T := B \cap N$ is normal in $N$, and the quotient $W := N/T$ admits a set of generators $S$ such that the following two conditions hold:

(BN1) For $s \in S$ and $w \in W$, we have $sBw \subseteq BswB \cup BwB$.

(BN2) For $s \in S$, we have $sBs^{-1} \nleq B$.

A group $G$ with a BN-pair indeed has a Bruhat decomposition with $C(w) := BwB$, and hence has an associated building which we can call $\Delta(G)$. The Solomon-Tits theorem holds in this generality:

Theorem (Solomon-Tits). Suppose $(G,B,N,W,S)$ is a group with a BN-pair whose Coxeter system is finite with longest element $w_0$, and write $q_0 := |B : B \cap w_0 B w_0^{-1}|$. Then the homotopy type of the associated building has the wedge decomposition $$\Delta(G) \simeq \bigvee_{i=1}^{q_0} \large{\mathbb{S}}^{|S|-1} \, .$$

We have not incorporated the prime $p$ into the definition of a BN-pair yet. That is the next step.

Definition 3. A BN-pair $(G,B,N,W,S)$ is called split of characteristic $p$ if $G$ is finite, and

(SBN1) $T := B \cap N$ equals $B \cap w_0 B w_0^{-1}$, and is abelian of order prime to $p$.

(SBN2) $B$ splits as a semi-direct product $B = U \rtimes T$ with $U$ a $p$-group.

If moreover $U \cap w_I U w_I^{-1}$ is normal in $U$ for all $I \subseteq S$ (here $w_I$ is the longest element of $W_I := \langle I \rangle_{W}$), then the BN-pair is called strongly split.

Writing $U_I := U \cap w_I U w_I^{-1}$ above, Cabanes-Enguehard show in their book that in the strongly split situation we have $N_G(U_I) = P_I$ and $U_I = O_p(P_I)$. Thus each $U_I$ is a $p$-radical subgroup of $G$, and we get a well-defined map $$\Phi: \Delta(G) \rightarrow \mathcal{B}_p(G)$$ defined by $\Phi(gP_I) := gU_Ig^{-1}$.

Now if $gP_J \subseteq hP_I$, then $P_J \subseteq P_I$ and $gP_I = hP_I$, so it follows from elementary results that $gU_Ig^{-1} \subseteq hU_Jh^{-1}$, and vice versa.

We see that, keeping in mind that the definition of $\Delta(G)$ has reverse inclusions, $\Phi$ is an order preserving map which identifies $\Delta(G)$ with its image in $\mathcal{B}_p(G)$. The obvious question is whether $\Phi$ is surjective. The answer is yes for the "Lie type" case:

Theorem (Borel-Tits). Let $\mathbf{G}$ be a simple linear algebraic group defined over an algebraically closed field of characteristic $p \neq 0$. Let $\sigma$ be an endomorphism of $\mathbf{G}$ onto itself such that the $\sigma$-fixed points $G:= \mathbf{G}^{\sigma}$ form a finite group. Then $\Phi$ is surjective.

This amounts to the last corollary of Burgoyne-Williamson's more elementary paper. The endomorphism setup is from Steinberg's work, who proved a Bruhat decomposition under these assumptions, although I couldn't find the 1968 memoirs of Steinberg online. I hope I am not misrepresenting the result, say, due to conflicting use of the term "parabolic" in these works. Also, I think Borel-Tits' original work allows to generalize to the reductive case.

Note, in particular, we now have a proof for Quillen's result I quoted in the beginning (Quillen's own argument is more direct but requires similar input): First note $\mathcal{A}_p(G) \simeq \mathcal{B}_p(G)$ and then Borel-Tits says $\mathcal{B}_p(G)$ is actually isomorphic with $\Delta(G)$ as a poset.

In Chapter 6 of their book, Cabanes-Enguehard tackle Alperin's weight conjecture for a finite group with a strongly split BN-pair of characteristic $p$, in the defining characteristic. The proof is given together with the extra assumption that $\Phi$ is surjective, suggesting a negative answer to my first question, in which case the second question becomes interesting:

Let $G$ be a finite group with a strongly split BN-pair of characteristic $p$.

Question 1. Is $\Phi$ necessarily surjective?

Question 2. Does $\Phi$ necessarily induce a homotopy equivalence?


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