Hausdorffen dimentsioa eta kutxa-dimentsioa geometria fraktalaren esparruko oinarrizko kontzeptuak dira. Horiek ohiko dimentsio topologikoaren ideia orokortzen dute; izan ere, objektu geometriko arrunten dimentsioari ohiko balio bera ematen dieten bitartean, intuitiboki hain dimentsio argia ez duten zenbait multzo patologikori dimentsio ez oso bat esleitzen diote. Bestalde, azken hamarkadetan, Hausdorffen dimentsioaren kontzeptuak aplikazio emankor eta interesgarriak eman ditu oinarri zenbakigarriko talde profinituen testuinguruan, talde hauei espazio metriko egitura eman baitakieke. Artikulu honetan, alde batetik, Hausdorffen dimentsioaren eta kutxa-dimentsioaren teoriaren sarrera orokor bat emango dugu. Beste aldetik, Hausdorffen dimentsioak talde profinituetan eman dituen zenbait emaitza esanguratsu azalduko dira, arreta bi alor konkretutara bideratuz: Hausdorffen espektroa eta talde R-analitikoak.
We prove that every word is strongly concise in the class of compact R-analytic groups.
IntroductionA word in k-variables is an element w(x 1 , . . . , x k ) of the free group F (x 1 , . . . , x k ), and, given any group G, it inherently defines a map w : G (k) → G. The image of that map, which will be denoted by w{G}, is the set of word-values of w, and it is typically not a subgroup of G. However, we can naturally associate two subgroups to w: the verbal subgroup defined as w(G) = w{G} , and the marginal subgroup, that is,A word is said to be concise in G if w{G} being finite implies that w(G) is finite. More generally, w is concise in a class C of groups if it is concise in all the groups G ∈ C, and w is simply called concise if it is concise in any group. P. Hall initially conjectured that all words were concise, but Ivanov [8] proved that conjecture to be false. Nevertheless, as Jaikin-Zapirain highlighted in [11], P. Hall's conjecture is still open for profinite groups: Profinite Conciseness Conjecture. Every word is concise in the class of profinite groups.
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