Let $G$ be a (rank $\geq 2$) semisimple real Lie group or a product of automorphism groups of trees. Then $G$ has fixed price 1, meaning the cost of every free pmp action is $1$, answering a question of Gaboriau for this class of groups. Moreover, this implies that such $G$ have vanishing rank gradient, settling a conjecture of Abért-Gelander-Nikolov. I am reading this paper with Petr and Alek. The most difficult part of this paper for me is the necessary background on higher-rank real semisimple Lie groups, structure theory of symmetric spaces, Busemann functions etc. However, Petr and Alek both know this machinery well. **Note (Oct 11th, 2026):** This reading project is on temporary hiatus. **Authors**: Mikołaj Frączyk, Sam Mellick, and Amanda Wilkens. **Year**: Received 2023, accepted 2025. **Journal**: *Annals of Mathematics*. **Paper link**: [ArXiv](https://arxiv.org/pdf/2307.01194). **Number of pages**: 44. **Reading this with**: [Petr Kosenko](https://sites.google.com/view/pkosenko/home), a postdoc at University of Haifa starting Oct 2026 and [Aleksander Skenderi](https://sites.google.com/view/aleksanderskenderi/home), a postdoc at Jagellonian working with Mikołaj. > [! Tentative Schedule] > - **Week 1** (Petr) Gentle introduction to symmetric spaces and the differences between rank 1 and higher-rank spaces. > - **Week 2** (Petr) Section 4: Computations for real semisimple Lie groups. Cartan decomposition, Weyl group, positive restricted root system. Computations with $G = SL_n(\mathbb{R})$. Langlands decomposition. > - **Week 3** (Alek) Background on classification of Lie groups and Lie algebras. Root systems, positive restricted root systems. > - **Week 4** (Alek) More background and computations with $SL(2, \mathbb{R})$ and $SL(3, \mathbb{R})$. Visual boundary. > - **Week 4** (Tasmin) Theory of countable Borel equivalence relations, cost of countable groups. Point processes as a way to calculate the cost of continuum-size locally compact second countable groups. > - **Week 5** (Petr) Background on point processes and Poisson point processes, Mecke equation, Palm calculus. [[30Sep_PPP.pdf|Petr's notes.]]