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In this note we establish some general finiteness results concerning lattices Γ in connected Lie groups$G$which possess certain “density” properties (see$Moskowitz$, M., On the density theorems of Borel and Furstenberg,$Ark. Mat.$$16$(1978), 11–27, and$Moskowitz$, M., Some results on automorphisms of bounded displacement and bounded cocycles,$Monatsh. Math.$$85$(1978), 323–336). For such groups we show that Γ always has finite index in its normalizer$N$_{$G$}(Γ). We then investigate analogous questions for the automorphism group Aut($G$) proving, under appropriate conditions, that Stab_{Aut($G$)}(Γ) is discrete. Finally we show, under appropriate conditions, that the subgroup $\tilde{\Gamma}=\{i_{\gamma}:\gamma \in \Gamma \},\ i_{\gamma}(x)=\gamma x\gamma^{-1}$ , of Aut($G$) has finite index in Stab_{Aut($G$)}(Γ). We test the limits of our results with various examples and counterexamples.

In this thesis, we consider some aspects of$noncommutative classical invariant theory$, i.e., noncommutative invariants of$the classical group SL(2, k)$. We develop a$symbolic method$for invariants and covariants, and we use the method to compute some invariant algebras. The subspace$Ĩ$_{d}^{m}of the noncommutative invariant algebra$Ĩ$_{$d$}consisting of homogeneous elements of degree$m$has the structure of a module over the$symmetric group S$_{$m$}. We find the explicit decomposition into irreducible modules. As a consequence, we obtain the$Hilbert series$of the commutative classical invariant algebras. The$Cayley—Sylvester theorem$and the$Hermite reciprocity law$are studied in some detail. We consider a new power series H($Ĩ$_{d},$t$) whose coefficients are the number of irreducible$S$_{$m$}-modules in the decomposition of$Ĩ$_{d}^{m}, and show that it is rational. Finally, we develop some analogues of all this for covariants.

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