Topological restrictions on Anosov representations

We characterize groups admitting Anosov representations into $\mathsf{SL}(3,\mathbb R)$, projective Anosov representations into $\mathsf{SL}(4,\mathbb R)$, and Borel Anosov representations into $\mathsf{SL}(4,\mathbb R)$. More generally, we obtain bounds on the cohomological dimension of groups admitting $P_k$-Anosov representations into $\mathsf{SL}(d,\mathbb R)$ and offer several characterizations of Benoist representations

Anosov representations, strongly convex cocompact groups and weak eigenvalue gaps

We provide characterizations of Anosov representations of word hyperbolic groups into real semisimple Lie groups in terms of equivariant limit maps, the Cartan property and the uniform gap summation property of \cite{GGKW}. As an application we obtain a characterization of strongly convex cocompact subgroups of the projective linear group $\mathsf{PGL}(d, \mathbb{R})$. We also compute the H\"older exponent of the Anosov limit maps of an Anosov representation in terms of the Cartan and Lyapunov projection of the image.Comment: 45 pages, Comments welcom

The H\"older exponent of Anosov limit maps

Let $\Gamma$ be a non-elementary word hyperbolic group and $d_{a}, a>1,$ a visual metric on its Gromov boundary $\partial_{\infty}\Gamma$. For an $1$-Anosov representation $\rho:\Gamma \rightarrow \mathsf{GL}_{d}(\mathbb{K})$, where $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$, we calculate the H\"older exponent of the Anosov limit map $\xi_{\rho}^1:(\partial_{\infty}\Gamma, d_{a})\rightarrow (\mathbb{P}(\mathbb{K}^d),d_{\mathbb{P}})$ of $\rho$ in terms of the moduli of eigenvalues of elements in $\rho(\Gamma)$ and the stable translation length on $\Gamma$. If $\rho$ is either irreducible or $\xi_{\rho}^1(\partial_{\infty}\Gamma)$ spans $\mathbb{K}^d$ and $\rho$ is $\{1,2\}$-Anosov, then $\xi_{\rho}^1$ attains its H\"older exponent. We also provide an analogous calculation for the exponent of the inverse limit map of $(1,1,2)$-hyperconvex representations. Finally, we exhibit examples of non semisimple $1$-Anosov representations of surface groups in $\mathsf{SL}_4(\mathbb{R})$ whose Anosov limit map in $\mathbb{P}(\mathbb{R}^4)$ does not attain its H\"older exponent.Comment: 24 page

Federated Fine-Tuning of Foundation Models via Probabilistic Masking

Foundation Models (FMs) have revolutionized machine learning with their adaptability and high performance across tasks; yet, their integration into Federated Learning (FL) is challenging due to substantial communication overhead from their extensive parameterization. Current communication-efficient FL strategies, such as gradient compression, reduce bitrates to around $1$ bit-per-parameter (bpp). However, these approaches fail to harness the characteristics of FMs, with their large number of parameters still posing a challenge to communication efficiency, even at these bitrate regimes. In this work, we present DeltaMask, a novel method that efficiently fine-tunes FMs in FL at an ultra-low bitrate, well below 1 bpp. DeltaMask employs stochastic masking to detect highly effective subnetworks within FMs and leverage stochasticity and sparsity in client masks to compress updates into a compact grayscale image using probabilistic filters, deviating from traditional weight training approaches. Our comprehensive evaluations across various datasets and architectures demonstrate DeltaMask efficiently achieves bitrates as low as 0.09 bpp, enhancing communication efficiency while maintaining FMs performance, as measured on 8 datasets and 5 pre-trained models of various network architectures.Comment: 19 pages, 9 figure

New nonlinear hyperbolic groups

We construct nonlinear hyperbolic groups which are large, torsion‐free, one‐ended, and admit a finite K(π,1). Our examples are built from superrigid cocompact rank one lattices via amalgamated free products and HNN extensions.Peer Reviewedhttps://deepblue.lib.umich.edu/bitstream/2027.42/149490/1/blms12248.pdfhttps://deepblue.lib.umich.edu/bitstream/2027.42/149490/2/blms12248_am.pd