# Functoriality for compactifications of locally symmetric spaces

Let $$X$$ be a symmetric space associated to an algebraic group $$G$$ defined over $$\mathbb{Q}$$ and $$G(\mathbb{R})$$ acts on $$X$$ from the left. Let $$\Gamma \subset G(\mathbb{Q})$$ be an arithmetic subgroup which we assume to be torsion-free(even neat). Then the locally symmetric space $$X_{\Gamma} \mathrel{:=} \Gamma \backslash X$$ is known to have several Satake compactifications depending on the rank of the group $$G$$. It is my understanding that the Satake compactification come from a geometrically rational representation often denoted by $$\tau$$. Note that this representation may itself not be rational. I am mostly interested in the following two types of compactifications.

1. When the representation $$\tau$$ is generic and $$\operatorname{\mathbb{Q}-rank}(G) = \operatorname{\mathbb{R}-rank}(G)$$, then the associated Satake compactification is same as the rBS (reductive Borel–Serre) compactification.

2. When the symmetric space is Hermitian and the associated locally symmetric space admits a minimal Satake compactification or Satake–Baily–Borel compactification.

Let us mention for clarity what we mean by "functoriality of rBS or minimal compactification". Let $$H$$ be an algebraic group defined over $$\mathbb{Q}$$ and $$\varphi : H \rightarrow G$$ be a morphism of groups with finite kernel(may even assume $$\varphi$$ is injective). Associated to this morphism, we get a map of symmetric spaces $$X_H \rightarrow X$$, where $$X_H$$ is a symmetric space associated to the group $$H$$. Let $$\Gamma_H \mathrel{:=} \varphi^{-1}(\Gamma)$$ and $$X_{\Gamma_H} \mathrel{:=} \Gamma_H \backslash X_H$$. We get a map of locally symmetric spaces $$\varphi : X_{\Gamma_H} \rightarrow X_{\Gamma}$$ induced by $$\varphi$$. We say a compactification is functorial if such a map $$\varphi$$ extends to a continuous map among the compactifications.

It is known that when $$X_{\Gamma_H}$$ and $$X_{\Gamma}$$ are Hermitian locally symmetric space then minimal compactification exists and it is functorial. It is also known that the rBS compactification is not functorial in general. I have the following questions.

Q1. Are there instances when rBS compactification is functorial? For example when $$G$$ and $$H$$ are of $$\mathbb{Q}$$-rank 1? or when $$\operatorname{\mathbb{Q}-rank}(H) = \operatorname{\mathbb{Q}-rank}(G)$$ and the root datum with respect to the maximal $$\mathbb{Q}$$-split tori are the same? or some other instance?

Q2. What is known about the functoriality of other Satake compactifications?

Q3. Let us denote by $$\hat{X}$$, a uniform way of compactifying locally symmetric spaces for example the ones described in 1 or 2. What can be said about the closure $$\overline{\varphi{X_{\Gamma_H}}} \subset \hat{X}$$? How is it related to $$\hat{X_{\Gamma_H}}$$? This is motivated by a remark in the book "Compactifications of symmetric and locally symmetric spaces" by A. Borel and L. Ji.

Q4. The rBS is not functorial but (perhaps surprisingly) the cohomology is in certain cases. This follows from one of the main results of A. Nair which describes a deRham model for the cohomology of rBS compactification of a Hermitian locally symmetric space. This appeared in "Weighted cohomology of arithmetic groups" by A. Nair. I wonder if there are ways to produce maps on the level of homotopy groups as well?

I apologize in advance for asking too many questions under the disguise of one. But they all seemed to belong together. I would appreciate any response towards a better understanding of any of these questions.

• What does "$\varphi : H \to G$ a morphism of groups with finite kernel or even a subgroup" mean? Do you mean "even the inclusion of a subgroup"? This seems to be a special case of a morphism with finite kernel, so I'm not sure why it would need to be singled out. – LSpice Mar 25 at 13:28
• @LSpice You are right. That is what I meant. I wanted to write that one may even consider this case only where $\varphi$ is an injective homomorphism. – random123 Mar 25 at 17:01
• About Q4, it's possible that such a question can be answered using Mikala Ørsnes Jansen's recent "combinatorial" model for the homotopy type of rBS, see arxiv.org/abs/2012.10777 . In which situations do Nair's results imply that the cohomology is functorial? – Dustin Clausen Mar 26 at 15:07
• @DustinClausen Thank you! The result cited in Q4 is proven to be true by Nair, when $X_{\Gamma}$ is a Hermitian locally symmetric space. The result you mention is certainly quite interesting. In the Jansen's combinatorial model for rBS, I am having trouble understanding how to define the map on objects. – random123 Mar 26 at 19:16
• Right, thanks. What I meant to ask, though, is for which maps $H\rightarrow G$ does Nair's result imply that the cohomology pulls back? I agree it's not obvious how to define the map on objects in Jansen's model (though perhaps this depends on $H\rightarrow G$ which is why I asked), but then again there's a bit more freedom if you only want a map on the homotopy level: it could for example be implemented by a span of maps where the first one is an equivalence. – Dustin Clausen Mar 26 at 20:30