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1. On the Automorphism Groups of Hyperbolic Manifolds

   https://doi.org/10.1093/imrn/rnaf083  

   Budney, Ryan; Gabai, David (April 2025, International Mathematics Research Notices)

   Abstract Let $${\mathrm{Diff}}_{0}(N)$$ represent the subgroup of diffeomorphisms that are homotopic to the identity. We show that if $$N$$ is a closed hyperbolic 4-manifold, then $$\pi _{0}{\mathrm{Diff}}_{0}(N)$$ is not finitely generated with similar results holding topologically. This proves in dimension-4 results previously known for $$n$$-dimensional hyperbolic manifolds of dimension $$n\ge 11$$ by Farrell and Jones in 1989 and $$n\ge 10$$ by Farrell and Ontaneda in 2010. Our proof relies on the technical result that $$\pi _{0}{\mathrm{Homeo}}(S^{1}\times D^{3})$$ is not finitely generated, which extends to the topological category smooth results of the authors. We also show that $$\pi _{n-4} {\mathrm{Homeo}}(S^{1} \times D^{n-1})$$ is not finitely generated for $$n \geq 4$$ and in particular $$\pi _{0}{\mathrm{Homeo}}(S^{1}\times D^{3})$$ is not finitely generated. These results are new for $n=4, 5$ and $$7$$. We also introduce higher dimensional barbell maps and establish some of their basic properties. 

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2. Nonnegative Ricci curvature, almost stability at infinity, and structure of fundamental groups

   https://doi.org/10.1016/j.aim.2025.110310  

   Pan, Jiayin (July 2025, Advances in Mathematics)

   We study the fundamental group of an open $$n$$-manifold $$M$$ of nonnegative Ricci curvature with additional stability conditions on $$\widetilde{M}$$, the Riemannian universal cover of $$M$$. We prove that if every asymptotic cone of $$\widetilde{M}$$ is a metric cone, whose cross-section is sufficiently Gromov-Hausdorff close to a prior fixed metric cone, then $$\pi_1(M)$$ is finitely generated and contains a normal abelian subgroup of finite index; if in addition $$\widetilde{M}$$ has Euclidean volume growth of constant at least $$L$$, then we can bound the index of that abelian subgroup by a constant $C(n,L)$. In particular, our result implies that if $$\widetilde{M}$$ has Euclidean volume growth of constant at least $$1-\epsilon(n)$$, then $$\pi_1(M)$$ is finitely generated and $C(n)$-abelian. 

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3. Finite-dimensional approximations of certain amalgamated free products of groups

   https://doi.org/10.4171/GGD/826  

   Schafhauser, Christopher (November 2024, Groups, Geometry, and Dynamics)

   A group is called matricial field (MF) if it admits finite-dimensional approximate unitary representations which are approximately faithful and approximately contained in the left regular representation. This paper provides a new class of MF groups by showing that given two amenable groups with a common normal subgroup, the amalgamated free product is MF. 

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4. Convex co‐compact representations of 3‐manifold groups

   https://doi.org/10.1112/topo.12332  

   Islam, Mitul; Zimmer, Andrew (May 2024, Journal of Topology)

   Abstract A representation of a finitely generated group into the projective general linear group is called convex co‐compact if it has finite kernel and its image acts convex co‐compactly on a properly convex domain in real projective space. We prove that the fundamental group of a closed irreducible orientable 3‐manifold can admit such a representation only when the manifold is geometric (with Euclidean, Hyperbolic or Euclidean Hyperbolic geometry) or when every component in the geometric decomposition is hyperbolic. In each case, we describe the structure of such examples. 

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5. Words, Hausdorff dimension and randomly free groups

   https://doi.org/https://doi.org/10.1007/s00208-017-1635-y  

   Larsen, Michael; Shalev, Aner (July 2018, Mathematische Annalen)

   We bound the size of fibers of word maps in finite and residually finite groups, and derive various applications. Our main result shows that, for any word $$1 \ne w \in F_d$$ there exists $$\e > 0$$ such that if $$\Gamma$$ is a residually finite group with infinitely many non-isomorphic non-abelian upper composition factors, then all fibers of the word map $$w:\Gamma^d \rightarrow \Gamma$$ have Hausdorff dimension at most $$d -\e$$. We conclude that profinite groups $$G := \hat\Gamma$$, $$\Gamma$$ as above, satisfy no probabilistic identity, and therefore they are \emph{randomly free}, namely, for any $$d \ge 1$$, the probability that randomly chosen elements $$g_1, \ldots , g_d \in G$$ freely generate a free subgroup (isomorphic to $$F_d$$) is $$1$$. This solves an open problem from \cite{DPSS}. Additional applications and related results are also established. For example, combining our results with recent results of Bors, we conclude that a profinite group in which the set of elements of finite odd order has positive measure has an open prosolvable subgroup. This may be regarded as a probabilistic version of the Feit-Thompson theorem. 

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