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This is an expository book on unitary representations of topological groups, and of several dual spaces, which are spaces of such representations up to some equivalence. The most important notions are defined for topological groups, but a special attention is paid to the case of discrete groups. The unitary dual of a group $G$ is the space of equivalence classes of its irreducible unitary representations; it is both a topological space and a Borel space. The primitive dual is the space of weak equivalence classes of unitary irreducible representations. The normal quasi-dual is the space of quasi-equivalence classes of traceable factor representations; it is parametrized by characters, which can be finite or infinite. The theory is systematically illustrated by a series of specific examples: Heisenberg groups, affine groups of infinite fields, solvable Baumslag-Solitar groups, lamplighter groups, and general linear groups. Operator algebras play an important role in the exposition, in particular the von Neumann algebras associated to a unitary representation and C*-algebras associated to a locally compact group.
Let $G$ be a group. A subset $F subset G$ is called irreducibly faithful if there exists an irreducible unitary representation $pi$ of $G$ such that $pi(x) eq mathrm{id}$ for all $x in F smallsetminus {e}$. Otherwise $F$ is called irreducibly unfait
We prove a generalization of a conjecture of C. Marion on generation properties of finite groups of Lie type, by considering geometric properties of an appropriate representation variety and associated tangent spaces.
We show that for a fixed k, Gromov random groups with any positive density have no non-trivial degree-k representations over any field, a.a.s. This is especially interesting in light of the results of Agol, Ollivier and Wise that when the density is
The degree pattern of a finite group is the degree sequence of its prime graph in ascending order of vertices. We say that the problem of OD-characterization is solved for a finite group if we determine the number of pairwise nonisomorphic finite gro
In this paper we prove that the unitary groups $SU_n(q^2)$ are $(2,3)$-generated for any prime power $q$ and any integer $ngeq 8$. By previous results this implies that, if $ngeq 3$, the groups $SU_n(q^2)$ and $PSU_n(q^2)$ are $(2,3)$-generated, exce