Questions in Geometric Group Theory — Q 10.1

Definitions and mathematical statement

Let $X$ be a finite-dimensional simply connected Riemannian symmetric space with nonpositive sectional curvature, and let $I=\operatorname{Isom}(X)$ with its usual Lie-group topology. For a finitely generated group $\Gamma$, put \[ D_X(\Gamma)=\{\rho:\Gamma\to I:\rho\text{ is injective and } \rho(\Gamma)\text{ is discrete}\}/I, \] where $I$ acts by conjugation. Give the homomorphism space the topology of pointwise convergence, equivalently convergence of the images of a fixed finite generating set, and give $D_X(\Gamma)$ the quotient topology.

Find algebraic or geometric conditions on $\Gamma$ that characterize or imply compactness of this deformation space, and corresponding conditions that ensure noncompactness. The ambient symmetric space is fixed while representations vary. Discreteness is a property of the image subgroup, while faithfulness requires trivial kernel; both must be retained in any proposed criterion.

Conjugation removes changes caused solely by moving the base frame in $X$. A useful compactness criterion therefore must control genuine deformation and degeneration after conjugation, including whether limiting representations remain discrete and faithful. The empty space is compact in the topological sense, so existence of at least one discrete faithful representation should be addressed separately when giving a substantive criterion. The source asks for conditions, not a universal assertion that all such spaces are compact or all are noncompact.

Short English statement

Which groups have compact spaces of discrete faithful representations into a fixed nonpositively curved symmetric space?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 1101001 (AMR-010-1001), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Bestvina - Questions in Geometric Group Theory (pdf) (2004), Question 10.1 (PDF page 18). Original source indexed by AMR-010-1001. https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf.