BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:David Handelman (University of Ottawa) DTSTART:20260128T200000Z DTEND:20260128T210000Z DTSTAMP:20260420T053211Z UID:NYC-NCG/218 DESCRIPTION:Title: Random walks on groups from a dimension group perspective\nby David Handelman (University of Ottawa) as part of Noncommutative geometry in NYC \n\n\nAbstract\nLet G be a finitely generated infinite discrete group\, an d let S\, containing 1\, be a finite\nsubset of G that generates it as a s emigroup (that is\, $U_{n=0}^{\\infty}S^n = G$). Let P be an element of\nt he group algebra AG (where A is either the integers or the reals)\, whose support is S\, and\nall of whose nonzero coefficients are positive. Then l eft multiplication by P is a positive\nhomomorphism $AG \\to AG$\, and ite rating it leads to an unnormalized random walk on G.\nWe can associate in the obvious way the structure of a dimension group (a direct limit of\nsim plicially ordered torsion-free abelian groups/finite-dimensional vector sp aces).\n\nWe are interested in space-time cones associated to this constru ction\, and the harmonic\nfunctions thereon (generalizing from the case of abelian groups\, a method of proving even-\ntual positivity for repeated multiplication by P)\, that reflect properties of the random\nwalk. A natu ral cone arises by setting $L_n$ to be the subset of G that can be reached by n\niterates of S starting at 1\, i.e.\, $L_n = S^n$\, and this has the advantage that at each stage\, we\nare dealing with finite-dimensional ve ctor spaces. However\, this is still quite complicated\nand massively redu ndant\; so we define Ln to be Sn with all points reached in fewer than n\n steps deleted. This is better from the dimension group point of view\, but there is now the\npossibility of dead-ends\, that is\, g in $L_n$ with $g · S\\subset   S^n$ (so no gs—with s in S—belongs\nto $L_{n+1} · S $)\, and these occur almost ubiquitously.\n\nWe first describe how we can refine $L_n$ to avoid dead-ends without loss of information\,\nand then st udy properties of the random walk that are naturally suggested by behaviou r\nof these new (almost-) partitions of G. Then we apply them to torsion-f ree abelian by\nfinite groups\, and show that some are much better behaved than others\, by considering\nthe induced integral action. Then we discus s other groups\, and in some cases\, determine\nthe pure (= extremal = ind ecomposable = ergodic) unfaithful finite harmonic functions on\nthem\, in particular\, for the simplest discrete Heisenberg group and the lamplighte r group.\nFinally\, we show that the quotients by the maximal order ideals of the resulting dimension\ngroups are always ordinary stationery dimensi on groups (and if A is the integers\, every\nsuch can be obtained for some free group and choice of P)\, so in particular\, have unique\ntrace. In t he case of the lamplighter group\, this exhausts the unfaithful pure harmo nic\nfunctions\, but in the case of the Heisenberg group\, don’t even am ount to a hill of traces.\n LOCATION:https://researchseminars.org/talk/NYC-NCG/218/ END:VEVENT END:VCALENDAR