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It is shown that all torsion-free cocompact lattices in any SL$(n,\mathbb{C})$ produce the desired integrality of the trace at least on the group ring, under a very weak Spin$^c$-assumption, in my thesis. Few, if any, of these groups are known to satisfy Baum-Connes for $n\ge 3$. I tend to think none of these are known for $n\ge 3$, but am not familiar enough with V.Lafforgues work. For SL$(2, \mathbb{C})$, any discrete subgroup satisfies Baum-Connes shown by Julg and Kasparov, as it acts on hyperbolic space.

The Kadison-Kaplansky-conjecture, in turn, was known much earlier for these groups; this goes back to Hyman Bass. Moreover, I think that the integrality on $\ell^1 \Gamma$ for many lattices follows from the proof of the Bost conjecture, due to V.Lafforgue.

For the $C^\ast$-algebra, the question is completely open in the cases not covered by Baum-Connes, at least to my knowledge.

The idea of the integrality proof is to carry out a careful version of the Dirac-dual Dirac construction, smooth enough for (a variant of) cyclic homology, and then resort to the integrality of an elliptic operator's index, as you said. Maybe this answers part of your question.

Perhaps one can say, therefore, that trace integrality is rather a problem in geometry than in representation theory (at least insofar as the group ring is concerned).

By the way, finitely generated free groups satisfy Baum-Connes; for instance because they are Gromov-hyperbolic, however, one can, as far as I remember, directly understand the right hand-side and find generators of the K-theory, see Bruce Blackadar's book.

You can read my thesis here: http://arxiv.org/abs/math/0612023 .

It is shown that all torsion-free cocompact lattices in any SL$(n,\mathbb{C})$ produce the desired integrality of the trace at least on the group ring, under a very weak Spin$^c$-assumption, in my thesis. (More precisely, the trace on the algebraic K-theory of the group ring with coefficients in the ring of rapidly decaying matrices is shown to be integer-valued, and this implies Kadison-Kaplansky for the group ring.) Few, if any, of these groups are known to satisfy Baum-Connes for $n\ge 3$. I think none of these are known for $n\ge 4$, and for $n=3$ they are known by the work of V. Lafforgue. For SL$(2, \mathbb{C})$, any discrete subgroup satisfies Baum-Connes shown by Julg and Kasparov, as it acts on hyperbolic space.

The Kadison-Kaplansky-conjecture, in turn, was known much earlier for these groups; this goes back to Hyman Bass. Moreover, I think that the integrality on $\ell^1 \Gamma$ for many lattices follows from the proof of the Bost conjecture, due to V. Lafforgue.

Therefore, the answer depends strongly on the type of algebra you require the trace to be defined on. For the $C^\ast$-algebra, the question is completely open in the cases not covered by Baum-Connes, at least to my knowledge (and I would be excited to hear to the contrary!)

The idea of the integrality proof in my thesis is related to the standard Baum-Connes proof: it is a careful version of the Dirac-dual Dirac construction, smooth enough for (a variant of) cyclic homology, and then resort to the integrality of an elliptic operator's index, as you said. Maybe this answers part of your question.

Perhaps one can say, therefore, that since all known proofs of trace integrality are geometric-analytic in nature, the problem can be viewed as a geometrical one as opposed to a representation theoretical one (at least insofar as the group ring is concerned).

By the way, finitely generated free groups satisfy Baum-Connes; for instance because they have the rapid decay property (or are the archtypical Gromov-hyperbolic groups), however, one can, as far as I remember, directly understand the right hand-side and write down generators of the K-theory, see Bruce Blackadar's book.

You can read my thesis here: http://arxiv.org/abs/math/0612023 .

It is shown that all torsion-free cocompact lattices in any SL$(n,\mathbb{C})$ produce the desired integrality of the trace at least on the group ring, under a very weak Spin$^c$-assumption, in my thesis. Few, if any, of these groups are known to satisfy Baum-Connes for $n\ge 3$. I tend to think none of these are known for $n\ge 3$, but am not familiar enough with V.Lafforgues work. For SL$(2, \mathbb{C})$, any discrete subgroup satisfies Baum-Connes shown by Julg and Kasparov, as it acts on hyerbolic space.

The Kadison-Kaplansky-conjecture, in turn, was known much earlier for these groups; this goes back to Hyman Bass. Moreover, I think that the integrality on $\ell^1 \Gamma$ for many lattices also follows from the proof of the Bost conjecture, due to V.Lafforgue.

For the $C^\ast$-algebra, the question is completely open in the cases not covered by Baum-Connes, at least to my knowledge.

The idea of the integrality proof is to carry out a careful version of the Dirac-dual Dirac construction, smooth enough for (a variant of) cyclic homology, and then resort to the integrality of an elliptic operator's index, as you said. Maybe this answers part of your question.

Perhaps one can say, therefore, that trace integrality is rather a problem in geometry than in representation theory (at least insofar as the group ring is concerned).

You can read my thesis here: http://arxiv.org/abs/math/0612023

By the way, finitely generated free groups satisfy Baum-Connes; for instance because they are Gromov-hyperbolic, however, one can, as far as I remember, directly understand the right hand-side and find generators of the K-theory, see Bruce Blackadar's book.

It is shown that all torsion-free cocompact lattices in any SL$(n,\mathbb{C})$ produce the desired integrality of the trace at least on the group ring, under a very weak Spin$^c$-assumption, in my thesis. Few, if any, of these groups are known to satisfy Baum-Connes for $n\ge 3$. I tend to think none of these are known for $n\ge 3$, but am not familiar enough with V.Lafforgues work. For SL$(2, \mathbb{C})$, any discrete subgroup satisfies Baum-Connes shown by Julg and Kasparov, as it acts on hyperbolic space.

The Kadison-Kaplansky-conjecture, in turn, was known much earlier for these groups; this goes back to Hyman Bass. Moreover, I think that the integrality on $\ell^1 \Gamma$ for many lattices follows from the proof of the Bost conjecture, due to V.Lafforgue.

For the $C^\ast$-algebra, the question is completely open in the cases not covered by Baum-Connes, at least to my knowledge.

The idea of the integrality proof is to carry out a careful version of the Dirac-dual Dirac construction, smooth enough for (a variant of) cyclic homology, and then resort to the integrality of an elliptic operator's index, as you said. Maybe this answers part of your question.

Perhaps one can say, therefore, that trace integrality is rather a problem in geometry than in representation theory (at least insofar as the group ring is concerned).

By the way, finitely generated free groups satisfy Baum-Connes; for instance because they are Gromov-hyperbolic, however, one can, as far as I remember, directly understand the right hand-side and find generators of the K-theory, see Bruce Blackadar's book.

You can read my thesis here: http://arxiv.org/abs/math/0612023 .

It is shown that all torsion-free cocompact lattices in any SL$(n,\mathbb{C})$ produce the desired integrality of the trace under a very weak Spin$^c$-assumption, in my thesis. Few, if any, of these groups are known to satisfy Baum-Connes for $n\ge 3$. (I tend to think none of these are known for $n\ge 3$, but am not familiar enough with V.Lafforgues work. For SL$(2, \mathbb{C})$, any discrete subgroup satisfies Baum-Connes shown by Julg and Kasparov, as it acts on hyerbolic space.)

The Kadison-Kaplansky-conjecture, in turn, was known much earlier for these groups; this goes back to Hyman Bass.

The idea of the integrality proof is to carry out a careful version of the Dirac-dual Dirac construction, smooth enough for (a variant of) cyclic homology.

Maybe one can say, therefore, that trace integrality is rather a problem in geometry than in representation theory.

You can read my thesis here: http://arxiv.org/abs/math/0612023

By the way, finitely generated free groups satisfy Baum-Connes; for instance, because they are Gromov-hyperbolic, but one can more or less directly understand the right hand-side and find generators.

It is shown that all torsion-free cocompact lattices in any SL$(n,\mathbb{C})$ produce the desired integrality of the trace at least on the group ring, under a very weak Spin$^c$-assumption, in my thesis. Few, if any, of these groups are known to satisfy Baum-Connes for $n\ge 3$. I tend to think none of these are known for $n\ge 3$, but am not familiar enough with V.Lafforgues work. For SL$(2, \mathbb{C})$, any discrete subgroup satisfies Baum-Connes shown by Julg and Kasparov, as it acts on hyerbolic space.

The Kadison-Kaplansky-conjecture, in turn, was known much earlier for these groups; this goes back to Hyman Bass. Moreover, I think that the integrality on $\ell^1 \Gamma$ for many lattices also follows from the proof of the Bost conjecture, due to V.Lafforgue.

For the $C^\ast$-algebra, the question is completely open in the cases not covered by Baum-Connes, at least to my knowledge.

The idea of the integrality proof is to carry out a careful version of the Dirac-dual Dirac construction, smooth enough for (a variant of) cyclic homology, and then resort to the integrality of an elliptic operator's index, as you said. Maybe this answers part of your question.

Perhaps one can say, therefore, that trace integrality is rather a problem in geometry than in representation theory (at least insofar as the group ring is concerned).

You can read my thesis here: http://arxiv.org/abs/math/0612023

By the way, finitely generated free groups satisfy Baum-Connes; for instance because they are Gromov-hyperbolic, however, one can, as far as I remember, directly understand the right hand-side and find generators of the K-theory, see Bruce Blackadar's book.