subgroup theorem
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2020 ◽  
pp. 1-30
Author(s):  
YVES BENOIST

Abstract The topic of this course is the discrete subgroups of semisimple Lie groups. We discuss a criterion that ensures that such a subgroup is arithmetic. This criterion is a joint work with Sébastien Miquel, which extends previous work of Selberg and Hee Oh and solves an old conjecture of Margulis. We focus on concrete examples like the group $\mathrm {SL}(d,{\mathbb {R}})$ and we explain how classical tools and new techniques enter the proof: the Auslander projection theorem, the Bruhat decomposition, the Mahler compactness criterion, the Borel density theorem, the Borel–Harish-Chandra finiteness theorem, the Howe–Moore mixing theorem, the Dani–Margulis recurrence theorem, the Raghunathan–Venkataramana finite-index subgroup theorem and so on.


2018 ◽  
Vol 28 (04) ◽  
pp. 605-611
Author(s):  
Tomasz Prytuła

Given a group [Formula: see text] with bounded torsion that acts properly on a systolic complex, we show that every solvable subgroup of [Formula: see text] is finitely generated and virtually abelian of rank at most [Formula: see text]. In particular, this gives a new proof of the above theorem for systolic groups. The main tools used in the proof are the Product Decomposition Theorem and the Flat Torus Theorem.


Author(s):  
Matt Clay

This chapter studies subgroups of free groups using the combinatorics of graphs and a simple operation called folding. It introduces a topological model for free groups and uses this model to show the rank of the free group H and whether every finitely generated nontrivial normal subgroup of a free group has finite index. The edge paths and the fundamental group of a graph are discussed, along with subgroups via graphs. The chapter also considers five applications of folding: the Nielsen–Schreier Subgroup theorem, the membership problem, index, normality, and residual finiteness. A group G is residually finite if for every nontrivial element g of G there is a normal subgroup N of finite index in G so that g is not in N. Exercises and research projects are included.


2016 ◽  
Vol 51 (2) ◽  
pp. 335-343
Author(s):  
Clemens Fuchs ◽  
◽  
Duc Hiep Pham ◽  

2016 ◽  
Vol 10 (3) ◽  
pp. 867-883 ◽  
Author(s):  
Richard Gaelan Hanlon ◽  
Eduardo Martínez Pedroza
Keyword(s):  

2014 ◽  
Vol 8 (3) ◽  
pp. 789-810 ◽  
Author(s):  
Darren Creutz ◽  
Yehuda Shalom

2013 ◽  
Vol 57 (1) ◽  
pp. 31-53
Author(s):  
Fedor Bogomolov ◽  
Frederick P. Greenleaf

AbstractWe provide an alternative constructive proof of the classical Brauer theorem for finite groups based on the well-known description of the complex irreducible representations of the symmetric groups Sn. The theorem is first proved for Sn and then for general G by embedding in Sn and applying the Mackey subgroup theorem.


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