Are there finitely presented infinite groups with a finite class number?
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1$\begingroup$ Do you mean number of conjugacy classes? $\endgroup$– Benjamin SteinbergCommented Mar 28, 2012 at 21:12
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1$\begingroup$ This appears as a problem here: sci.ccny.cuny.edu/~shpil/gworld/problems/probFP.html (Assuming that the class number is the number of conjugacy classes) $\endgroup$– Alessandro SistoCommented Mar 28, 2012 at 22:20
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$\begingroup$ According to Mark Sapir's answer to Derek Holt's question in math.niu.edu/~rusin/known-math/95/finite.conj, for each large prime p there exists a 2-generated infinite group of exponent p which has exactly p conjugacy classes (that's Theorem 41.2 in Olshansky, "Geometry of defining relations in groups"). $\endgroup$– RalphCommented Mar 28, 2012 at 23:10
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$\begingroup$ @Benjamin Steinberg: yes! The class number is the number of conjugacy classes of a group. $\endgroup$– Joerg SixtCommented Apr 2, 2012 at 10:48
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Denis Osin proved that there was a finitely generated infinite group with exactly two (count'em) conjugacy classes, but I can't seem to find any statement as to the finiteness of the presentation. (see this paper by Ashot Minasian for references and more results).
answered Mar 28, 2012 at 23:18
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1$\begingroup$ @Igor: Osin's group is infinitely presented, which is also true for Ivanov's examples cited above. No finitely presented examples are known and there are no methods which can produce such examples even "theoretically". It is quite possible that there are no finitely presented examples at all. $\endgroup$– user6976Commented Mar 29, 2012 at 0:33
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$\begingroup$ @Mark: thanks! I think I first saw Denis' examples because such a group (if I remember correctly) would not have any finite-dimensional unitary representations, but I guess it is much easier to not have such representations than to have few conjugacy classes... $\endgroup$ Commented Mar 29, 2012 at 1:12
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3$\begingroup$ Finitely presented "monsters" are discussed on pages 3 and 4 here: arxiv.org/pdf/1103.3873.pdf $\endgroup$– user6976Commented Mar 29, 2012 at 1:39