| Let R be {dollar}doubc{lcub}lambda{rcub}, doubclbracklbracklambdarbrackrbrack,{dollar} or more generally a discrete valuation ring of characteristic zero whose residue field is algebraically closed. For such a ring, let {dollar}Vsb{lcub}R{rcub}{dollar} denote a free R-module of rank n, and let {dollar}{lcub}cal N{rcub}(Vsb{lcub}R{rcub}{dollar}) denote the nilpotent endomorphisms of {dollar}Vsb{lcub}R{rcub}, {lcub}cal N{rcub}(Vsb{lcub}R{rcub})subseteq gl(Vsb{lcub}R{rcub}){dollar}. Elements of {dollar}GL(Vsb{lcub}R{rcub}){dollar} act on {dollar}{lcub}cal N{rcub}(Vsb{lcub}R{rcub}){dollar} by conjugation, and this thesis is an attempt to determine the structure of {dollar}GL(Vsb{lcub}R{rcub}){lcub}cal N{rcub}(Vsb{lcub}R{rcub}).{dollar} We determine a canonical set of discrete invariants for an orbit in {dollar}GL(Vsb{lcub}R{rcub}{lcub}cal N{rcub}(Vsb{lcub}R{rcub}),{dollar} and use these and other invariants to establish congruence bounds for conjugacy relations in {dollar}GL(Vsb{lcub}R{rcub}){lcub}cal N{rcub}(Vsb{lcub}R{rcub}).{dollar} These results are then applied to questions of moduli for the orbit space {dollar}GL(Vsb{lcub}R{rcub}){lcub}cal N{rcub}(Vsb{lcub}R{rcub}),{dollar} where the finite dimensionality of subsets of orbits with a fixed set of discrete invariants is established, under additional assumptions, for instance when {dollar}R = doubclbracklbracklambdarbrackrbrack .{dollar}... |
