On homomorphisms of crossed products of locally indicable groups to - - PowerPoint PPT Presentation

On homomorphisms of crossed products of locally indicable groups to division rings

Andrei Jaikin-Zapirain UAM and ICMAT Spa, June 2019

Andrei Jaikin-Zapirain UAM and ICMAT Homomorphisms to division rings

Embeddings of domains into division rings

A commutative domain R can be embedded in a field R ֒ → Q(R) = {ab−1 : a ∈ R, 0 = b ∈ R}. Question Can we embed a noncommutative domain R into a division ring? The (right) Ore condition: aR ∩ bR = {0} if a, b = 0. The Ore condition ⇒ R ֒ → Q(R) = {ab−1 : a ∈ R, 0 = b ∈ R}. A Noetherian domain is embedded into a division ring.

• A. I. Malcev (1937): There exists a noncomutative domain that

cannot be embedded in a division ring.

• A. J. Bowtell’s example(1967):

Andrei Jaikin-Zapirain UAM and ICMAT Homomorphisms to division rings

Embeddings of domains into division rings

A commutative domain R can be embedded in a field R ֒ → Q(R) = {ab−1 : a ∈ R, 0 = b ∈ R}. Question Can we embed a noncommutative domain R into a division ring? The (right) Ore condition: aR ∩ bR = {0} if a, b = 0. The Ore condition ⇒ R ֒ → Q(R) = {ab−1 : a ∈ R, 0 = b ∈ R}. A Noetherian domain is embedded into a division ring.

• A. I. Malcev (1937): There exists a noncomutative domain that

cannot be embedded in a division ring.

• A. J. Bowtell’s example(1967):

Andrei Jaikin-Zapirain UAM and ICMAT Homomorphisms to division rings

Embeddings of domains into division rings

A commutative domain R can be embedded in a field R ֒ → Q(R) = {ab−1 : a ∈ R, 0 = b ∈ R}. Question Can we embed a noncommutative domain R into a division ring? The (right) Ore condition: aR ∩ bR = {0} if a, b = 0. The Ore condition ⇒ R ֒ → Q(R) = {ab−1 : a ∈ R, 0 = b ∈ R}. A Noetherian domain is embedded into a division ring.

• A. I. Malcev (1937): There exists a noncomutative domain that

cannot be embedded in a division ring.

• A. J. Bowtell’s example(1967):

Andrei Jaikin-Zapirain UAM and ICMAT Homomorphisms to division rings

Embeddings of domains into division rings

A commutative domain R can be embedded in a field R ֒ → Q(R) = {ab−1 : a ∈ R, 0 = b ∈ R}. Question Can we embed a noncommutative domain R into a division ring? The (right) Ore condition: aR ∩ bR = {0} if a, b = 0. The Ore condition ⇒ R ֒ → Q(R) = {ab−1 : a ∈ R, 0 = b ∈ R}. A Noetherian domain is embedded into a division ring.

• A. I. Malcev (1937): There exists a noncomutative domain that

cannot be embedded in a division ring.

• A. J. Bowtell’s example(1967):

Andrei Jaikin-Zapirain UAM and ICMAT Homomorphisms to division rings

Embeddings of domains into division rings

A commutative domain R can be embedded in a field R ֒ → Q(R) = {ab−1 : a ∈ R, 0 = b ∈ R}. Question Can we embed a noncommutative domain R into a division ring? The (right) Ore condition: aR ∩ bR = {0} if a, b = 0. The Ore condition ⇒ R ֒ → Q(R) = {ab−1 : a ∈ R, 0 = b ∈ R}. A Noetherian domain is embedded into a division ring.

• A. I. Malcev (1937): There exists a noncomutative domain that

cannot be embedded in a division ring.

• A. J. Bowtell’s example(1967):

Andrei Jaikin-Zapirain UAM and ICMAT Homomorphisms to division rings

Embeddings of domains into division rings

A commutative domain R can be embedded in a field R ֒ → Q(R) = {ab−1 : a ∈ R, 0 = b ∈ R}. Question Can we embed a noncommutative domain R into a division ring? The (right) Ore condition: aR ∩ bR = {0} if a, b = 0. The Ore condition ⇒ R ֒ → Q(R) = {ab−1 : a ∈ R, 0 = b ∈ R}. A Noetherian domain is embedded into a division ring.

• A. I. Malcev (1937): There exists a noncomutative domain that

cannot be embedded in a division ring.

• A. J. Bowtell’s example(1967):

Andrei Jaikin-Zapirain UAM and ICMAT Homomorphisms to division rings

Embeddings of domains into division rings

A commutative domain R can be embedded in a field R ֒ → Q(R) = {ab−1 : a ∈ R, 0 = b ∈ R}. Question Can we embed a noncommutative domain R into a division ring? The (right) Ore condition: aR ∩ bR = {0} if a, b = 0. The Ore condition ⇒ R ֒ → Q(R) = {ab−1 : a ∈ R, 0 = b ∈ R}. A Noetherian domain is embedded into a division ring.

• A. I. Malcev (1937): There exists a noncomutative domain that

cannot be embedded in a division ring.

• A. J. Bowtell’s example(1967):

Andrei Jaikin-Zapirain UAM and ICMAT Homomorphisms to division rings

Embeddings of domains into division rings

A commutative domain R can be embedded in a field R ֒ → Q(R) = {ab−1 : a ∈ R, 0 = b ∈ R}. Question Can we embed a noncommutative domain R into a division ring? The (right) Ore condition: aR ∩ bR = {0} if a, b = 0. The Ore condition ⇒ R ֒ → Q(R) = {ab−1 : a ∈ R, 0 = b ∈ R}. A Noetherian domain is embedded into a division ring.

• A. I. Malcev (1937): There exists a noncomutative domain that

cannot be embedded in a division ring.

• A. J. Bowtell’s example(1967):

Andrei Jaikin-Zapirain UAM and ICMAT Homomorphisms to division rings

Embeddings of group rings into division rings

Can we embed a group ring K[G] or a crossed product E ∗ G into a division ring? In this talk K is always a field and E is a division algebra. When does K[G] have non-trivial zero-divisors? g − 1 is a zero-divisor if g ∈ G is of finite order. Kaplansky’s zero-divisor conjecture Assume G is torsion-free. Then K[G] and E ∗ G are domains.

• G. Higman (1940): if G is left-orderable, then K[G] (and E ∗ G)

are domains. Malcev’s conjecture, Problem 1.6 of the Kourovka notebook Assume G is left orderable. Then K[G] (and E ∗ G) can be embedded in a division ring.

Andrei Jaikin-Zapirain UAM and ICMAT Homomorphisms to division rings

Embeddings of group rings into division rings

Can we embed a group ring K[G] or a crossed product E ∗ G into a division ring? In this talk K is always a field and E is a division algebra. When does K[G] have non-trivial zero-divisors? g − 1 is a zero-divisor if g ∈ G is of finite order. Kaplansky’s zero-divisor conjecture Assume G is torsion-free. Then K[G] and E ∗ G are domains.

• G. Higman (1940): if G is left-orderable, then K[G] (and E ∗ G)

are domains. Malcev’s conjecture, Problem 1.6 of the Kourovka notebook Assume G is left orderable. Then K[G] (and E ∗ G) can be embedded in a division ring.

Andrei Jaikin-Zapirain UAM and ICMAT Homomorphisms to division rings

Embeddings of group rings into division rings

Can we embed a group ring K[G] or a crossed product E ∗ G into a division ring? In this talk K is always a field and E is a division algebra. When does K[G] have non-trivial zero-divisors? g − 1 is a zero-divisor if g ∈ G is of finite order. Kaplansky’s zero-divisor conjecture Assume G is torsion-free. Then K[G] and E ∗ G are domains.

• G. Higman (1940): if G is left-orderable, then K[G] (and E ∗ G)

are domains. Malcev’s conjecture, Problem 1.6 of the Kourovka notebook Assume G is left orderable. Then K[G] (and E ∗ G) can be embedded in a division ring.

Andrei Jaikin-Zapirain UAM and ICMAT Homomorphisms to division rings

Embeddings of group rings into division rings

Can we embed a group ring K[G] or a crossed product E ∗ G into a division ring? In this talk K is always a field and E is a division algebra. When does K[G] have non-trivial zero-divisors? g − 1 is a zero-divisor if g ∈ G is of finite order. Kaplansky’s zero-divisor conjecture Assume G is torsion-free. Then K[G] and E ∗ G are domains.

• G. Higman (1940): if G is left-orderable, then K[G] (and E ∗ G)

are domains. Malcev’s conjecture, Problem 1.6 of the Kourovka notebook Assume G is left orderable. Then K[G] (and E ∗ G) can be embedded in a division ring.

Andrei Jaikin-Zapirain UAM and ICMAT Homomorphisms to division rings

Embeddings of group rings into division rings

Can we embed a group ring K[G] or a crossed product E ∗ G into a division ring? In this talk K is always a field and E is a division algebra. When does K[G] have non-trivial zero-divisors? g − 1 is a zero-divisor if g ∈ G is of finite order. Kaplansky’s zero-divisor conjecture Assume G is torsion-free. Then K[G] and E ∗ G are domains.

• G. Higman (1940): if G is left-orderable, then K[G] (and E ∗ G)

are domains. Malcev’s conjecture, Problem 1.6 of the Kourovka notebook Assume G is left orderable. Then K[G] (and E ∗ G) can be embedded in a division ring.

Andrei Jaikin-Zapirain UAM and ICMAT Homomorphisms to division rings

Embeddings of group rings into division rings

Can we embed a group ring K[G] or a crossed product E ∗ G into a division ring? In this talk K is always a field and E is a division algebra. When does K[G] have non-trivial zero-divisors? g − 1 is a zero-divisor if g ∈ G is of finite order. Kaplansky’s zero-divisor conjecture Assume G is torsion-free. Then K[G] and E ∗ G are domains.

• G. Higman (1940): if G is left-orderable, then K[G] (and E ∗ G)

are domains. Malcev’s conjecture, Problem 1.6 of the Kourovka notebook Assume G is left orderable. Then K[G] (and E ∗ G) can be embedded in a division ring.

Andrei Jaikin-Zapirain UAM and ICMAT Homomorphisms to division rings

Embeddings of group rings into division rings

Can we embed a group ring K[G] or a crossed product E ∗ G into a division ring? In this talk K is always a field and E is a division algebra. When does K[G] have non-trivial zero-divisors? g − 1 is a zero-divisor if g ∈ G is of finite order. Kaplansky’s zero-divisor conjecture Assume G is torsion-free. Then K[G] and E ∗ G are domains.

• G. Higman (1940): if G is left-orderable, then K[G] (and E ∗ G)

are domains. Malcev’s conjecture, Problem 1.6 of the Kourovka notebook Assume G is left orderable. Then K[G] (and E ∗ G) can be embedded in a division ring.

Andrei Jaikin-Zapirain UAM and ICMAT Homomorphisms to division rings

Embeddings of crossed products into division rings

Malcev’s conjecture, Problem 1.6 of the Kourovka notebook Assume that G is left-orderable. Then K[G] (and E ∗ G) can be embedded in a division ring. Theorem (A. I. Malcev (1948), B. H. Neumann (1949), H. Hahn (1907) (the abelian case)) Let be a bi-invariant order on G. Then the set K((G, )) of formal power series over G with coefficients in K having well-ordered support has a natural structure of a ring and, moreover, it is a division ring. The same holds for E ∗ G if G is bi-orderable. Let be a bi-invariant order on G. Then the set MN (E ∗ G) of formal infinite sums of homogeneous elements of E ∗ G having well-ordered support has a natural structure of a ring and, moreover, it is a division ring.

Andrei Jaikin-Zapirain UAM and ICMAT Homomorphisms to division rings

Embeddings of crossed products into division rings

Malcev’s conjecture, Problem 1.6 of the Kourovka notebook Assume that G is left-orderable. Then K[G] (and E ∗ G) can be embedded in a division ring. Theorem (A. I. Malcev (1948), B. H. Neumann (1949), H. Hahn (1907) (the abelian case)) Let be a bi-invariant order on G. Then the set K((G, )) of formal power series over G with coefficients in K having well-ordered support has a natural structure of a ring and, moreover, it is a division ring. The same holds for E ∗ G if G is bi-orderable. Let be a bi-invariant order on G. Then the set MN (E ∗ G) of formal infinite sums of homogeneous elements of E ∗ G having well-ordered support has a natural structure of a ring and, moreover, it is a division ring.

Andrei Jaikin-Zapirain UAM and ICMAT Homomorphisms to division rings

Embeddings of crossed products into division rings

Malcev’s conjecture, Problem 1.6 of the Kourovka notebook Assume that G is left-orderable. Then K[G] (and E ∗ G) can be embedded in a division ring. Theorem (A. I. Malcev (1948), B. H. Neumann (1949), H. Hahn (1907) (the abelian case)) Let be a bi-invariant order on G. Then the set K((G, )) of formal power series over G with coefficients in K having well-ordered support has a natural structure of a ring and, moreover, it is a division ring. The same holds for E ∗ G if G is bi-orderable. Let be a bi-invariant order on G. Then the set MN (E ∗ G) of formal infinite sums of homogeneous elements of E ∗ G having well-ordered support has a natural structure of a ring and, moreover, it is a division ring.

Andrei Jaikin-Zapirain UAM and ICMAT Homomorphisms to division rings

Embeddings of crossed product of locally indicable groups into division rings

A group G is called locally indicable if every non-trivial finitely generated group of G maps onto Z. Examples: free groups, surface groups, Thompson’s group F, torsion-free subgroups of one-relator groups, amenable left-orderable groups, bi-orderable groups. A locally indicable group is left-orderable.

• P. Conrad (1959): A group G is locally indicable if and only if G

has a Conradian order. Theorem A (A. Jaikin (2019)) Let G be a locally indicable group and let E be a division ring. Then E ∗ G can be embedded in a division ring.

Andrei Jaikin-Zapirain UAM and ICMAT Homomorphisms to division rings

Embeddings of crossed product of locally indicable groups into division rings

A group G is called locally indicable if every non-trivial finitely generated group of G maps onto Z. Examples: free groups, surface groups, Thompson’s group F, torsion-free subgroups of one-relator groups, amenable left-orderable groups, bi-orderable groups. A locally indicable group is left-orderable.

• P. Conrad (1959): A group G is locally indicable if and only if G

has a Conradian order. Theorem A (A. Jaikin (2019)) Let G be a locally indicable group and let E be a division ring. Then E ∗ G can be embedded in a division ring.

Andrei Jaikin-Zapirain UAM and ICMAT Homomorphisms to division rings

Embeddings of crossed product of locally indicable groups into division rings

A group G is called locally indicable if every non-trivial finitely generated group of G maps onto Z. Examples: free groups, surface groups, Thompson’s group F, torsion-free subgroups of one-relator groups, amenable left-orderable groups, bi-orderable groups. A locally indicable group is left-orderable.

• P. Conrad (1959): A group G is locally indicable if and only if G

has a Conradian order. Theorem A (A. Jaikin (2019)) Let G be a locally indicable group and let E be a division ring. Then E ∗ G can be embedded in a division ring.

Andrei Jaikin-Zapirain UAM and ICMAT Homomorphisms to division rings

Embeddings of crossed product of locally indicable groups into division rings

A group G is called locally indicable if every non-trivial finitely generated group of G maps onto Z. Examples: free groups, surface groups, Thompson’s group F, torsion-free subgroups of one-relator groups, amenable left-orderable groups, bi-orderable groups. A locally indicable group is left-orderable.

• P. Conrad (1959): A group G is locally indicable if and only if G

has a Conradian order. Theorem A (A. Jaikin (2019)) Let G be a locally indicable group and let E be a division ring. Then E ∗ G can be embedded in a division ring.

Andrei Jaikin-Zapirain UAM and ICMAT Homomorphisms to division rings

Embeddings of crossed product of locally indicable groups into division rings

A group G is called locally indicable if every non-trivial finitely generated group of G maps onto Z. Examples: free groups, surface groups, Thompson’s group F, torsion-free subgroups of one-relator groups, amenable left-orderable groups, bi-orderable groups. A locally indicable group is left-orderable.

• P. Conrad (1959): A group G is locally indicable if and only if G

has a Conradian order. Theorem A (A. Jaikin (2019)) Let G be a locally indicable group and let E be a division ring. Then E ∗ G can be embedded in a division ring.

Andrei Jaikin-Zapirain UAM and ICMAT Homomorphisms to division rings

The proof of Theorem A: Dubrovin’s idea

For an arbitrary left-invariant order , MN (E ∗ G) is not a ring, but a left E ∗ G-module. Let E(E ∗ G) be the ring of endomorphisms of the abelian group MN (E ∗ G). Thus, E ∗ G ֒ → E(E ∗ G) . Conjecture (N. I. Dubrovin (1987)) The division closure D(E ∗ G) of E ∗ G in E(E ∗ G) is a division ring.

• N. I. Dubrovin (1987), G. M. Bergman (2018): The non-zero

elements of E ∗ G are invertible in E(E ∗ G). Theorem A (continuation) Let be a Conradian order on G. Then D(E ∗ G) is a division ring.

Andrei Jaikin-Zapirain UAM and ICMAT Homomorphisms to division rings

The proof of Theorem A: Dubrovin’s idea

For an arbitrary left-invariant order , MN (E ∗ G) is not a ring, but a left E ∗ G-module. Let E(E ∗ G) be the ring of endomorphisms of the abelian group MN (E ∗ G). Thus, E ∗ G ֒ → E(E ∗ G) . Conjecture (N. I. Dubrovin (1987)) The division closure D(E ∗ G) of E ∗ G in E(E ∗ G) is a division ring.

• N. I. Dubrovin (1987), G. M. Bergman (2018): The non-zero

elements of E ∗ G are invertible in E(E ∗ G). Theorem A (continuation) Let be a Conradian order on G. Then D(E ∗ G) is a division ring.

Andrei Jaikin-Zapirain UAM and ICMAT Homomorphisms to division rings

The proof of Theorem A: Dubrovin’s idea

For an arbitrary left-invariant order , MN (E ∗ G) is not a ring, but a left E ∗ G-module. Let E(E ∗ G) be the ring of endomorphisms of the abelian group MN (E ∗ G). Thus, E ∗ G ֒ → E(E ∗ G) . Conjecture (N. I. Dubrovin (1987)) The division closure D(E ∗ G) of E ∗ G in E(E ∗ G) is a division ring.

• N. I. Dubrovin (1987), G. M. Bergman (2018): The non-zero

elements of E ∗ G are invertible in E(E ∗ G). Theorem A (continuation) Let be a Conradian order on G. Then D(E ∗ G) is a division ring.

Andrei Jaikin-Zapirain UAM and ICMAT Homomorphisms to division rings

The proof of Theorem A: Dubrovin’s idea

For an arbitrary left-invariant order , MN (E ∗ G) is not a ring, but a left E ∗ G-module. Let E(E ∗ G) be the ring of endomorphisms of the abelian group MN (E ∗ G). Thus, E ∗ G ֒ → E(E ∗ G) . Conjecture (N. I. Dubrovin (1987)) The division closure D(E ∗ G) of E ∗ G in E(E ∗ G) is a division ring.

• N. I. Dubrovin (1987), G. M. Bergman (2018): The non-zero

elements of E ∗ G are invertible in E(E ∗ G). Theorem A (continuation) Let be a Conradian order on G. Then D(E ∗ G) is a division ring.

Andrei Jaikin-Zapirain UAM and ICMAT Homomorphisms to division rings

The proof of Theorem A: Dubrovin’s idea

For an arbitrary left-invariant order , MN (E ∗ G) is not a ring, but a left E ∗ G-module. Let E(E ∗ G) be the ring of endomorphisms of the abelian group MN (E ∗ G). Thus, E ∗ G ֒ → E(E ∗ G) . Conjecture (N. I. Dubrovin (1987)) The division closure D(E ∗ G) of E ∗ G in E(E ∗ G) is a division ring.

• N. I. Dubrovin (1987), G. M. Bergman (2018): The non-zero

elements of E ∗ G are invertible in E(E ∗ G). Theorem A (continuation) Let be a Conradian order on G. Then D(E ∗ G) is a division ring.

Andrei Jaikin-Zapirain UAM and ICMAT Homomorphisms to division rings

Hughes-free division E ∗ G-algebras

Are D(E ∗ G) isomorphic as E ∗ G-rings for different ? Theorem (I. Hughes (1970), W. Dicks, D. Herbera, J. S´ anchez (2004)) Let E ∗ G be a crossed product of a division ring E and a locally indicable group G. Then up to E ∗ G-isomorphism there exists

• nly one epic Hughes-free division E ∗ G-ring.

Notice that the existence of a Hughes-free division E ∗ G-algebra was known only in some particular cases, but not in general. Theorem A (continuation) Let be a Conradian order on G. Then D(E ∗ G) is a strongly Hughes-free division E ∗ G-ring. Therefore, D(E ∗ G) does not depend on a Conradian order . In the following we denote D(E ∗ G) by DE∗G.

Andrei Jaikin-Zapirain UAM and ICMAT Homomorphisms to division rings

Hughes-free division E ∗ G-algebras

Are D(E ∗ G) isomorphic as E ∗ G-rings for different ? Theorem (I. Hughes (1970), W. Dicks, D. Herbera, J. S´ anchez (2004)) Let E ∗ G be a crossed product of a division ring E and a locally indicable group G. Then up to E ∗ G-isomorphism there exists

• nly one epic Hughes-free division E ∗ G-ring.

Notice that the existence of a Hughes-free division E ∗ G-algebra was known only in some particular cases, but not in general. Theorem A (continuation) Let be a Conradian order on G. Then D(E ∗ G) is a strongly Hughes-free division E ∗ G-ring. Therefore, D(E ∗ G) does not depend on a Conradian order . In the following we denote D(E ∗ G) by DE∗G.

Andrei Jaikin-Zapirain UAM and ICMAT Homomorphisms to division rings

Hughes-free division E ∗ G-algebras

Are D(E ∗ G) isomorphic as E ∗ G-rings for different ? Theorem (I. Hughes (1970), W. Dicks, D. Herbera, J. S´ anchez (2004)) Let E ∗ G be a crossed product of a division ring E and a locally indicable group G. Then up to E ∗ G-isomorphism there exists

• nly one epic Hughes-free division E ∗ G-ring.

Notice that the existence of a Hughes-free division E ∗ G-algebra was known only in some particular cases, but not in general. Theorem A (continuation) Let be a Conradian order on G. Then D(E ∗ G) is a strongly Hughes-free division E ∗ G-ring. Therefore, D(E ∗ G) does not depend on a Conradian order . In the following we denote D(E ∗ G) by DE∗G.

Andrei Jaikin-Zapirain UAM and ICMAT Homomorphisms to division rings

Hughes-free division E ∗ G-algebras

Are D(E ∗ G) isomorphic as E ∗ G-rings for different ? Theorem (I. Hughes (1970), W. Dicks, D. Herbera, J. S´ anchez (2004)) Let E ∗ G be a crossed product of a division ring E and a locally indicable group G. Then up to E ∗ G-isomorphism there exists

• nly one epic Hughes-free division E ∗ G-ring.

Notice that the existence of a Hughes-free division E ∗ G-algebra was known only in some particular cases, but not in general. Theorem A (continuation) Let be a Conradian order on G. Then D(E ∗ G) is a strongly Hughes-free division E ∗ G-ring. Therefore, D(E ∗ G) does not depend on a Conradian order . In the following we denote D(E ∗ G) by DE∗G.

Andrei Jaikin-Zapirain UAM and ICMAT Homomorphisms to division rings

Hughes-free division E ∗ G-algebras

Are D(E ∗ G) isomorphic as E ∗ G-rings for different ? Theorem (I. Hughes (1970), W. Dicks, D. Herbera, J. S´ anchez (2004)) Let E ∗ G be a crossed product of a division ring E and a locally indicable group G. Then up to E ∗ G-isomorphism there exists

• nly one epic Hughes-free division E ∗ G-ring.

Notice that the existence of a Hughes-free division E ∗ G-algebra was known only in some particular cases, but not in general. Theorem A (continuation) Let be a Conradian order on G. Then D(E ∗ G) is a strongly Hughes-free division E ∗ G-ring. Therefore, D(E ∗ G) does not depend on a Conradian order . In the following we denote D(E ∗ G) by DE∗G.

Andrei Jaikin-Zapirain UAM and ICMAT Homomorphisms to division rings

Hughes-free division E ∗ G-algebras

Are D(E ∗ G) isomorphic as E ∗ G-rings for different ? Theorem (I. Hughes (1970), W. Dicks, D. Herbera, J. S´ anchez (2004)) Let E ∗ G be a crossed product of a division ring E and a locally indicable group G. Then up to E ∗ G-isomorphism there exists

• nly one epic Hughes-free division E ∗ G-ring.

Notice that the existence of a Hughes-free division E ∗ G-algebra was known only in some particular cases, but not in general. Theorem A (continuation) Let be a Conradian order on G. Then D(E ∗ G) is a strongly Hughes-free division E ∗ G-ring. Therefore, D(E ∗ G) does not depend on a Conradian order . In the following we denote D(E ∗ G) by DE∗G.

Andrei Jaikin-Zapirain UAM and ICMAT Homomorphisms to division rings

The universal division R-ring

A division R-ring ϕ : R → D induces a Sylvester matrix rank function rkD: for A ∈ Matn×m(R) rkD(A) := the D-rank of ϕ(A).

• P. M. Cohn (1971): two division R-rings ϕ : R → D and

ψ : R → E are isomorphic if and only if rkD = rkE. An epic division R-ring ϕ : R → D is universal if for every division R-ring ψ : R → E and every matrix A over R, rkD(A) ≥ rkE(A). If, moreover, ϕ is injective, D is called universal skew field of fractions. If R is a commutative domain, then R ֒ → Q(R) is universal.

• D. Passman (1982): there is a Noetherian domain R that does not

have the universal division R-ring.

• A. Jaikin (2019): if the group algebra Q[G] has the universal skew

field of fractions, then G is locally indicable.

Andrei Jaikin-Zapirain UAM and ICMAT Homomorphisms to division rings

The universal division R-ring

A division R-ring ϕ : R → D induces a Sylvester matrix rank function rkD: for A ∈ Matn×m(R) rkD(A) := the D-rank of ϕ(A).

• P. M. Cohn (1971): two division R-rings ϕ : R → D and

ψ : R → E are isomorphic if and only if rkD = rkE. An epic division R-ring ϕ : R → D is universal if for every division R-ring ψ : R → E and every matrix A over R, rkD(A) ≥ rkE(A). If, moreover, ϕ is injective, D is called universal skew field of fractions. If R is a commutative domain, then R ֒ → Q(R) is universal.

• D. Passman (1982): there is a Noetherian domain R that does not

have the universal division R-ring.

• A. Jaikin (2019): if the group algebra Q[G] has the universal skew

field of fractions, then G is locally indicable.

Andrei Jaikin-Zapirain UAM and ICMAT Homomorphisms to division rings

The universal division R-ring

A division R-ring ϕ : R → D induces a Sylvester matrix rank function rkD: for A ∈ Matn×m(R) rkD(A) := the D-rank of ϕ(A).

• P. M. Cohn (1971): two division R-rings ϕ : R → D and

ψ : R → E are isomorphic if and only if rkD = rkE. An epic division R-ring ϕ : R → D is universal if for every division R-ring ψ : R → E and every matrix A over R, rkD(A) ≥ rkE(A). If, moreover, ϕ is injective, D is called universal skew field of fractions. If R is a commutative domain, then R ֒ → Q(R) is universal.

• D. Passman (1982): there is a Noetherian domain R that does not

have the universal division R-ring.

• A. Jaikin (2019): if the group algebra Q[G] has the universal skew

field of fractions, then G is locally indicable.

Andrei Jaikin-Zapirain UAM and ICMAT Homomorphisms to division rings

The universal division R-ring

A division R-ring ϕ : R → D induces a Sylvester matrix rank function rkD: for A ∈ Matn×m(R) rkD(A) := the D-rank of ϕ(A).

• P. M. Cohn (1971): two division R-rings ϕ : R → D and

ψ : R → E are isomorphic if and only if rkD = rkE. An epic division R-ring ϕ : R → D is universal if for every division R-ring ψ : R → E and every matrix A over R, rkD(A) ≥ rkE(A). If, moreover, ϕ is injective, D is called universal skew field of fractions. If R is a commutative domain, then R ֒ → Q(R) is universal.

• D. Passman (1982): there is a Noetherian domain R that does not

have the universal division R-ring.

• A. Jaikin (2019): if the group algebra Q[G] has the universal skew

field of fractions, then G is locally indicable.

Andrei Jaikin-Zapirain UAM and ICMAT Homomorphisms to division rings

The universal division R-ring

A division R-ring ϕ : R → D induces a Sylvester matrix rank function rkD: for A ∈ Matn×m(R) rkD(A) := the D-rank of ϕ(A).

• P. M. Cohn (1971): two division R-rings ϕ : R → D and

ψ : R → E are isomorphic if and only if rkD = rkE. An epic division R-ring ϕ : R → D is universal if for every division R-ring ψ : R → E and every matrix A over R, rkD(A) ≥ rkE(A). If, moreover, ϕ is injective, D is called universal skew field of fractions. If R is a commutative domain, then R ֒ → Q(R) is universal.

• D. Passman (1982): there is a Noetherian domain R that does not

have the universal division R-ring.

• A. Jaikin (2019): if the group algebra Q[G] has the universal skew

field of fractions, then G is locally indicable.

Andrei Jaikin-Zapirain UAM and ICMAT Homomorphisms to division rings

The universal division R-ring

A division R-ring ϕ : R → D induces a Sylvester matrix rank function rkD: for A ∈ Matn×m(R) rkD(A) := the D-rank of ϕ(A).

• P. M. Cohn (1971): two division R-rings ϕ : R → D and

ψ : R → E are isomorphic if and only if rkD = rkE. An epic division R-ring ϕ : R → D is universal if for every division R-ring ψ : R → E and every matrix A over R, rkD(A) ≥ rkE(A). If, moreover, ϕ is injective, D is called universal skew field of fractions. If R is a commutative domain, then R ֒ → Q(R) is universal.

• D. Passman (1982): there is a Noetherian domain R that does not

have the universal division R-ring.

• A. Jaikin (2019): if the group algebra Q[G] has the universal skew

field of fractions, then G is locally indicable.

Andrei Jaikin-Zapirain UAM and ICMAT Homomorphisms to division rings

The universal division R-ring

A division R-ring ϕ : R → D induces a Sylvester matrix rank function rkD: for A ∈ Matn×m(R) rkD(A) := the D-rank of ϕ(A).

• P. M. Cohn (1971): two division R-rings ϕ : R → D and

ψ : R → E are isomorphic if and only if rkD = rkE. An epic division R-ring ϕ : R → D is universal if for every division R-ring ψ : R → E and every matrix A over R, rkD(A) ≥ rkE(A). If, moreover, ϕ is injective, D is called universal skew field of fractions. If R is a commutative domain, then R ֒ → Q(R) is universal.

• D. Passman (1982): there is a Noetherian domain R that does not

have the universal division R-ring.

• A. Jaikin (2019): if the group algebra Q[G] has the universal skew

field of fractions, then G is locally indicable.

Andrei Jaikin-Zapirain UAM and ICMAT Homomorphisms to division rings

The universal division E ∗ G-ring

Assume G is locally indicable. Is there the universal division E ∗ G-ring?

• A. Jaikin, D. L´
• pez-´

Alvarez (2018): Assume G is locally indicable. If there are the epic Hughes-free division E ∗ G-ring and the universal division E ∗ G-ring, then they are isomorphic. Assume G is locally indicable. Is DE∗G universal?

• J. Lewin (1974): DK[F] is universal if F is a free group.
• D. S. Passman (2009): DK[G] ∼

= Q(K[G]) is universal if G is a polycyclic group. A locally indicable group G is called Lewin if DE∗G is universal for any crossed product E ∗ G.

Andrei Jaikin-Zapirain UAM and ICMAT Homomorphisms to division rings

The universal division E ∗ G-ring

Assume G is locally indicable. Is there the universal division E ∗ G-ring?

• A. Jaikin, D. L´
• pez-´

Alvarez (2018): Assume G is locally indicable. If there are the epic Hughes-free division E ∗ G-ring and the universal division E ∗ G-ring, then they are isomorphic. Assume G is locally indicable. Is DE∗G universal?

• J. Lewin (1974): DK[F] is universal if F is a free group.
• D. S. Passman (2009): DK[G] ∼

= Q(K[G]) is universal if G is a polycyclic group. A locally indicable group G is called Lewin if DE∗G is universal for any crossed product E ∗ G.

Andrei Jaikin-Zapirain UAM and ICMAT Homomorphisms to division rings

The universal division E ∗ G-ring

Assume G is locally indicable. Is there the universal division E ∗ G-ring?

• A. Jaikin, D. L´
• pez-´

Alvarez (2018): Assume G is locally indicable. If there are the epic Hughes-free division E ∗ G-ring and the universal division E ∗ G-ring, then they are isomorphic. Assume G is locally indicable. Is DE∗G universal?

• J. Lewin (1974): DK[F] is universal if F is a free group.
• D. S. Passman (2009): DK[G] ∼

= Q(K[G]) is universal if G is a polycyclic group. A locally indicable group G is called Lewin if DE∗G is universal for any crossed product E ∗ G.

Andrei Jaikin-Zapirain UAM and ICMAT Homomorphisms to division rings

The universal division E ∗ G-ring

Assume G is locally indicable. Is there the universal division E ∗ G-ring?

• A. Jaikin, D. L´
• pez-´

Alvarez (2018): Assume G is locally indicable. If there are the epic Hughes-free division E ∗ G-ring and the universal division E ∗ G-ring, then they are isomorphic. Assume G is locally indicable. Is DE∗G universal?

• J. Lewin (1974): DK[F] is universal if F is a free group.
• D. S. Passman (2009): DK[G] ∼

= Q(K[G]) is universal if G is a polycyclic group. A locally indicable group G is called Lewin if DE∗G is universal for any crossed product E ∗ G.

Andrei Jaikin-Zapirain UAM and ICMAT Homomorphisms to division rings

The universal division E ∗ G-ring

Assume G is locally indicable. Is there the universal division E ∗ G-ring?

• A. Jaikin, D. L´
• pez-´

Alvarez (2018): Assume G is locally indicable. If there are the epic Hughes-free division E ∗ G-ring and the universal division E ∗ G-ring, then they are isomorphic. Assume G is locally indicable. Is DE∗G universal?

• J. Lewin (1974): DK[F] is universal if F is a free group.
• D. S. Passman (2009): DK[G] ∼

= Q(K[G]) is universal if G is a polycyclic group. A locally indicable group G is called Lewin if DE∗G is universal for any crossed product E ∗ G.

Andrei Jaikin-Zapirain UAM and ICMAT Homomorphisms to division rings

The universal division E ∗ G-ring

Assume G is locally indicable. Is there the universal division E ∗ G-ring?

• A. Jaikin, D. L´
• pez-´

Alvarez (2018): Assume G is locally indicable. If there are the epic Hughes-free division E ∗ G-ring and the universal division E ∗ G-ring, then they are isomorphic. Assume G is locally indicable. Is DE∗G universal?

• J. Lewin (1974): DK[F] is universal if F is a free group.
• D. S. Passman (2009): DK[G] ∼

= Q(K[G]) is universal if G is a polycyclic group. A locally indicable group G is called Lewin if DE∗G is universal for any crossed product E ∗ G.

Andrei Jaikin-Zapirain UAM and ICMAT Homomorphisms to division rings

Lewin groups

Theorem B (A. Jaikin (2019)) Let G be a locally indicable group.

1 If all finitely generated subgroups of G are Lewin, then G is

also Lewin.

2 Any subgroup of a Lewin group is also Lewin. 3 G is Lewin if G has a normal Lewin subgroup N such that

G/N is amenable and locally indicable.

4 Any limit in the space of n-generated marked groups of Lewin

groups is Lewin.

5 A direct product of Lewin groups is Lewin.

Corollary Amenable locally indicable groups, residually polycyclic groups, limit groups and free-by-cyclic groups are Lewin.

Andrei Jaikin-Zapirain UAM and ICMAT Homomorphisms to division rings

Lewin groups

Theorem B (A. Jaikin (2019)) Let G be a locally indicable group.

1 If all finitely generated subgroups of G are Lewin, then G is

also Lewin.

2 Any subgroup of a Lewin group is also Lewin. 3 G is Lewin if G has a normal Lewin subgroup N such that

G/N is amenable and locally indicable.

4 Any limit in the space of n-generated marked groups of Lewin

groups is Lewin.

5 A direct product of Lewin groups is Lewin.

Corollary Amenable locally indicable groups, residually polycyclic groups, limit groups and free-by-cyclic groups are Lewin.

Andrei Jaikin-Zapirain UAM and ICMAT Homomorphisms to division rings

Lewin groups

Theorem B (A. Jaikin (2019)) Let G be a locally indicable group.

1 If all finitely generated subgroups of G are Lewin, then G is

also Lewin.

2 Any subgroup of a Lewin group is also Lewin. 3 G is Lewin if G has a normal Lewin subgroup N such that

G/N is amenable and locally indicable.

4 Any limit in the space of n-generated marked groups of Lewin

groups is Lewin.

5 A direct product of Lewin groups is Lewin.

Corollary Amenable locally indicable groups, residually polycyclic groups, limit groups and free-by-cyclic groups are Lewin.

Andrei Jaikin-Zapirain UAM and ICMAT Homomorphisms to division rings

Lewin groups

Theorem B (A. Jaikin (2019)) Let G be a locally indicable group.

1 If all finitely generated subgroups of G are Lewin, then G is

also Lewin.

2 Any subgroup of a Lewin group is also Lewin. 3 G is Lewin if G has a normal Lewin subgroup N such that

G/N is amenable and locally indicable.

4 Any limit in the space of n-generated marked groups of Lewin

groups is Lewin.

5 A direct product of Lewin groups is Lewin.

Corollary Amenable locally indicable groups, residually polycyclic groups, limit groups and free-by-cyclic groups are Lewin.

Andrei Jaikin-Zapirain UAM and ICMAT Homomorphisms to division rings

Lewin groups

Theorem B (A. Jaikin (2019)) Let G be a locally indicable group.

1 If all finitely generated subgroups of G are Lewin, then G is

also Lewin.

2 Any subgroup of a Lewin group is also Lewin. 3 G is Lewin if G has a normal Lewin subgroup N such that

G/N is amenable and locally indicable.

4 Any limit in the space of n-generated marked groups of Lewin

groups is Lewin.

5 A direct product of Lewin groups is Lewin.

Corollary Amenable locally indicable groups, residually polycyclic groups, limit groups and free-by-cyclic groups are Lewin.

Andrei Jaikin-Zapirain UAM and ICMAT Homomorphisms to division rings

Lewin groups

Theorem B (A. Jaikin (2019)) Let G be a locally indicable group.

1 If all finitely generated subgroups of G are Lewin, then G is

also Lewin.

2 Any subgroup of a Lewin group is also Lewin. 3 G is Lewin if G has a normal Lewin subgroup N such that

G/N is amenable and locally indicable.

4 Any limit in the space of n-generated marked groups of Lewin

groups is Lewin.

5 A direct product of Lewin groups is Lewin.

Corollary Amenable locally indicable groups, residually polycyclic groups, limit groups and free-by-cyclic groups are Lewin.

Andrei Jaikin-Zapirain UAM and ICMAT Homomorphisms to division rings

Lewin groups

Theorem B (A. Jaikin (2019)) Let G be a locally indicable group.

1 If all finitely generated subgroups of G are Lewin, then G is

also Lewin.

2 Any subgroup of a Lewin group is also Lewin. 3 G is Lewin if G has a normal Lewin subgroup N such that

G/N is amenable and locally indicable.

4 Any limit in the space of n-generated marked groups of Lewin

groups is Lewin.

5 A direct product of Lewin groups is Lewin.

Corollary Amenable locally indicable groups, residually polycyclic groups, limit groups and free-by-cyclic groups are Lewin.

Andrei Jaikin-Zapirain UAM and ICMAT Homomorphisms to division rings

The embedding of C[G] into U(G)

Let G be a countable group. We denote by λ and ρ the left, and respectively, right regular representation of G in l2(G): λ(g)δh = δgh, ρ(s)δh = δhg−1. N(G) = the centralizer of λ(G) in the space of bounded operators

• n l2(G). N(G) is called the von Neumann group algebra of G.

N(G) satisfies the left and right Ore conditions. Its classical Ore ring of fractions is denoted by U(G) and its called the ring of unbounded affiliated operators of G. U(G) is a von Neumann regular ring: for ever a ∈ U(G) there exists b ∈ U(G) such that aba = a. Example: G = Z.

Andrei Jaikin-Zapirain UAM and ICMAT Homomorphisms to division rings

The embedding of C[G] into U(G)

Let G be a countable group. We denote by λ and ρ the left, and respectively, right regular representation of G in l2(G): λ(g)δh = δgh, ρ(s)δh = δhg−1. N(G) = the centralizer of λ(G) in the space of bounded operators

• n l2(G). N(G) is called the von Neumann group algebra of G.

N(G) satisfies the left and right Ore conditions. Its classical Ore ring of fractions is denoted by U(G) and its called the ring of unbounded affiliated operators of G. U(G) is a von Neumann regular ring: for ever a ∈ U(G) there exists b ∈ U(G) such that aba = a. Example: G = Z.

Andrei Jaikin-Zapirain UAM and ICMAT Homomorphisms to division rings

The embedding of C[G] into U(G)

Let G be a countable group. We denote by λ and ρ the left, and respectively, right regular representation of G in l2(G): λ(g)δh = δgh, ρ(s)δh = δhg−1. N(G) = the centralizer of λ(G) in the space of bounded operators

• n l2(G). N(G) is called the von Neumann group algebra of G.

N(G) satisfies the left and right Ore conditions. Its classical Ore ring of fractions is denoted by U(G) and its called the ring of unbounded affiliated operators of G. U(G) is a von Neumann regular ring: for ever a ∈ U(G) there exists b ∈ U(G) such that aba = a. Example: G = Z.

Andrei Jaikin-Zapirain UAM and ICMAT Homomorphisms to division rings

The embedding of C[G] into U(G)

Let G be a countable group. We denote by λ and ρ the left, and respectively, right regular representation of G in l2(G): λ(g)δh = δgh, ρ(s)δh = δhg−1. N(G) = the centralizer of λ(G) in the space of bounded operators

• n l2(G). N(G) is called the von Neumann group algebra of G.

N(G) satisfies the left and right Ore conditions. Its classical Ore ring of fractions is denoted by U(G) and its called the ring of unbounded affiliated operators of G. U(G) is a von Neumann regular ring: for ever a ∈ U(G) there exists b ∈ U(G) such that aba = a. Example: G = Z.

Andrei Jaikin-Zapirain UAM and ICMAT Homomorphisms to division rings

The embedding of C[G] into U(G)

Let G be a countable group. We denote by λ and ρ the left, and respectively, right regular representation of G in l2(G): λ(g)δh = δgh, ρ(s)δh = δhg−1. N(G) = the centralizer of λ(G) in the space of bounded operators

• n l2(G). N(G) is called the von Neumann group algebra of G.

N(G) satisfies the left and right Ore conditions. Its classical Ore ring of fractions is denoted by U(G) and its called the ring of unbounded affiliated operators of G. U(G) is a von Neumann regular ring: for ever a ∈ U(G) there exists b ∈ U(G) such that aba = a. Example: G = Z.

Andrei Jaikin-Zapirain UAM and ICMAT Homomorphisms to division rings

The embedding of C[G] into U(G)

Let G be a countable group. We denote by λ and ρ the left, and respectively, right regular representation of G in l2(G): λ(g)δh = δgh, ρ(s)δh = δhg−1. N(G) = the centralizer of λ(G) in the space of bounded operators

• n l2(G). N(G) is called the von Neumann group algebra of G.

N(G) satisfies the left and right Ore conditions. Its classical Ore ring of fractions is denoted by U(G) and its called the ring of unbounded affiliated operators of G. U(G) is a von Neumann regular ring: for ever a ∈ U(G) there exists b ∈ U(G) such that aba = a. Example: G = Z.

Andrei Jaikin-Zapirain UAM and ICMAT Homomorphisms to division rings

The embedding of C[G] into U(G)

Let G be a countable group. We denote by λ and ρ the left, and respectively, right regular representation of G in l2(G): λ(g)δh = δgh, ρ(s)δh = δhg−1. N(G) = the centralizer of λ(G) in the space of bounded operators

• n l2(G). N(G) is called the von Neumann group algebra of G.

N(G) satisfies the left and right Ore conditions. Its classical Ore ring of fractions is denoted by U(G) and its called the ring of unbounded affiliated operators of G. U(G) is a von Neumann regular ring: for ever a ∈ U(G) there exists b ∈ U(G) such that aba = a. Example: G = Z.

Andrei Jaikin-Zapirain UAM and ICMAT Homomorphisms to division rings

The embedding of C[G] into U(G)

Let G be a countable group. We denote by λ and ρ the left, and respectively, right regular representation of G in l2(G): λ(g)δh = δgh, ρ(s)δh = δhg−1. N(G) = the centralizer of λ(G) in the space of bounded operators

• n l2(G). N(G) is called the von Neumann group algebra of G.

N(G) satisfies the left and right Ore conditions. Its classical Ore ring of fractions is denoted by U(G) and its called the ring of unbounded affiliated operators of G. U(G) is a von Neumann regular ring: for ever a ∈ U(G) there exists b ∈ U(G) such that aba = a. Example: G = Z.

Andrei Jaikin-Zapirain UAM and ICMAT Homomorphisms to division rings

The strong Atiyah conjecture for torsion-free groups

The strong Atiyah conjecture for torsion-free groups Let G be a torsion-free group. Then the the division closure D(G)

• f ρ(C[G]) in U(G) is a division ring.

The standard formulation: rkG(A) ∈ Z, for any matrix A over C[G], where rkG is the von Neumann rank function. The conjecture implies Kaplansky’s conjecture for C[G]. It is known in many cases, where Kaplansky’s conjecture is known, but it is still open for left-orderable groups. Theorem C (A. Jaikin, D. L´

• pez-´

Alvarez (2018)) The strong Atiyah conjecture holds for locally indicable groups. Moreover, D(G) is Hughes-free. Thus, D(G) and DC[G] are isomorphic as C[G]-rings and rkG = rkDC[G].

Andrei Jaikin-Zapirain UAM and ICMAT Homomorphisms to division rings

The strong Atiyah conjecture for torsion-free groups

The strong Atiyah conjecture for torsion-free groups Let G be a torsion-free group. Then the the division closure D(G)

• f ρ(C[G]) in U(G) is a division ring.

The standard formulation: rkG(A) ∈ Z, for any matrix A over C[G], where rkG is the von Neumann rank function. The conjecture implies Kaplansky’s conjecture for C[G]. It is known in many cases, where Kaplansky’s conjecture is known, but it is still open for left-orderable groups. Theorem C (A. Jaikin, D. L´

• pez-´

Alvarez (2018)) The strong Atiyah conjecture holds for locally indicable groups. Moreover, D(G) is Hughes-free. Thus, D(G) and DC[G] are isomorphic as C[G]-rings and rkG = rkDC[G].

Andrei Jaikin-Zapirain UAM and ICMAT Homomorphisms to division rings

The strong Atiyah conjecture for torsion-free groups

The strong Atiyah conjecture for torsion-free groups Let G be a torsion-free group. Then the the division closure D(G)

• f ρ(C[G]) in U(G) is a division ring.

The standard formulation: rkG(A) ∈ Z, for any matrix A over C[G], where rkG is the von Neumann rank function. The conjecture implies Kaplansky’s conjecture for C[G]. It is known in many cases, where Kaplansky’s conjecture is known, but it is still open for left-orderable groups. Theorem C (A. Jaikin, D. L´

• pez-´

Alvarez (2018)) The strong Atiyah conjecture holds for locally indicable groups. Moreover, D(G) is Hughes-free. Thus, D(G) and DC[G] are isomorphic as C[G]-rings and rkG = rkDC[G].

Andrei Jaikin-Zapirain UAM and ICMAT Homomorphisms to division rings

The strong Atiyah conjecture for torsion-free groups

The strong Atiyah conjecture for torsion-free groups Let G be a torsion-free group. Then the the division closure D(G)

• f ρ(C[G]) in U(G) is a division ring.

The standard formulation: rkG(A) ∈ Z, for any matrix A over C[G], where rkG is the von Neumann rank function. The conjecture implies Kaplansky’s conjecture for C[G]. It is known in many cases, where Kaplansky’s conjecture is known, but it is still open for left-orderable groups. Theorem C (A. Jaikin, D. L´

• pez-´

Alvarez (2018)) The strong Atiyah conjecture holds for locally indicable groups. Moreover, D(G) is Hughes-free. Thus, D(G) and DC[G] are isomorphic as C[G]-rings and rkG = rkDC[G].

Andrei Jaikin-Zapirain UAM and ICMAT Homomorphisms to division rings

The strong Atiyah conjecture for torsion-free groups

The strong Atiyah conjecture for torsion-free groups Let G be a torsion-free group. Then the the division closure D(G)

• f ρ(C[G]) in U(G) is a division ring.

The standard formulation: rkG(A) ∈ Z, for any matrix A over C[G], where rkG is the von Neumann rank function. The conjecture implies Kaplansky’s conjecture for C[G]. It is known in many cases, where Kaplansky’s conjecture is known, but it is still open for left-orderable groups. Theorem C (A. Jaikin, D. L´

• pez-´

Alvarez (2018)) The strong Atiyah conjecture holds for locally indicable groups. Moreover, D(G) is Hughes-free. Thus, D(G) and DC[G] are isomorphic as C[G]-rings and rkG = rkDC[G].

Andrei Jaikin-Zapirain UAM and ICMAT Homomorphisms to division rings

The strong Atiyah conjecture for torsion-free groups

The strong Atiyah conjecture for torsion-free groups Let G be a torsion-free group. Then the the division closure D(G)

• f ρ(C[G]) in U(G) is a division ring.

The standard formulation: rkG(A) ∈ Z, for any matrix A over C[G], where rkG is the von Neumann rank function. The conjecture implies Kaplansky’s conjecture for C[G]. It is known in many cases, where Kaplansky’s conjecture is known, but it is still open for left-orderable groups. Theorem C (A. Jaikin, D. L´

• pez-´

Alvarez (2018)) The strong Atiyah conjecture holds for locally indicable groups. Moreover, D(G) is Hughes-free. Thus, D(G) and DC[G] are isomorphic as C[G]-rings and rkG = rkDC[G].

Andrei Jaikin-Zapirain UAM and ICMAT Homomorphisms to division rings

Parallelism between the properties of D(C[G]) and D(G)

C[G] ֒ → E(C[G]) C[G] ֒ → U(G) The division closure of C[G] D(C[G]) D(G) The elements of C[G] are

• N. I. Dubrovin
• P. Linnell

invertible if G is left-orderable (1987) (1992) The div. closure of C[G] is a Theorem A Theorem C

• div. ring if G is loc. indicable

(if is Conradian) Questions/Problems.

1 Let G be left-orderable. Are D(C[G]) and D(G) isomorphic? 2 Use analytic origin of D(G) ∼

= DC[G] in order to obtain new cases of loc. indicable groups G for which D(G) is universal.

3 Find new examples of locally indicable groups G such that

rkDC[G] = rkG(A) ≥ rkC(A) ∀ matrix A over Q[G] (rkC is induced by C[G] → C). Known: rkG(A) ≥ rkC(A) ∀ A over Q[G] ⇒ G is locally indicable.

Andrei Jaikin-Zapirain UAM and ICMAT Homomorphisms to division rings

Parallelism between the properties of D(C[G]) and D(G)

C[G] ֒ → E(C[G]) C[G] ֒ → U(G) The division closure of C[G] D(C[G]) D(G) The elements of C[G] are

• N. I. Dubrovin
• P. Linnell

invertible if G is left-orderable (1987) (1992) The div. closure of C[G] is a Theorem A Theorem C

• div. ring if G is loc. indicable

(if is Conradian) Questions/Problems.

1 Let G be left-orderable. Are D(C[G]) and D(G) isomorphic? 2 Use analytic origin of D(G) ∼

= DC[G] in order to obtain new cases of loc. indicable groups G for which D(G) is universal.

3 Find new examples of locally indicable groups G such that

rkDC[G] = rkG(A) ≥ rkC(A) ∀ matrix A over Q[G] (rkC is induced by C[G] → C). Known: rkG(A) ≥ rkC(A) ∀ A over Q[G] ⇒ G is locally indicable.

Andrei Jaikin-Zapirain UAM and ICMAT Homomorphisms to division rings

Parallelism between the properties of D(C[G]) and D(G)

C[G] ֒ → E(C[G]) C[G] ֒ → U(G) The division closure of C[G] D(C[G]) D(G) The elements of C[G] are

• N. I. Dubrovin
• P. Linnell

invertible if G is left-orderable (1987) (1992) The div. closure of C[G] is a Theorem A Theorem C

• div. ring if G is loc. indicable

(if is Conradian) Questions/Problems.

1 Let G be left-orderable. Are D(C[G]) and D(G) isomorphic? 2 Use analytic origin of D(G) ∼

= DC[G] in order to obtain new cases of loc. indicable groups G for which D(G) is universal.

3 Find new examples of locally indicable groups G such that

rkDC[G] = rkG(A) ≥ rkC(A) ∀ matrix A over Q[G] (rkC is induced by C[G] → C). Known: rkG(A) ≥ rkC(A) ∀ A over Q[G] ⇒ G is locally indicable.

Andrei Jaikin-Zapirain UAM and ICMAT Homomorphisms to division rings

Parallelism between the properties of D(C[G]) and D(G)

C[G] ֒ → E(C[G]) C[G] ֒ → U(G) The division closure of C[G] D(C[G]) D(G) The elements of C[G] are

• N. I. Dubrovin
• P. Linnell

invertible if G is left-orderable (1987) (1992) The div. closure of C[G] is a Theorem A Theorem C

• div. ring if G is loc. indicable

(if is Conradian) Questions/Problems.

1 Let G be left-orderable. Are D(C[G]) and D(G) isomorphic? 2 Use analytic origin of D(G) ∼

= DC[G] in order to obtain new cases of loc. indicable groups G for which D(G) is universal.

3 Find new examples of locally indicable groups G such that

rkDC[G] = rkG(A) ≥ rkC(A) ∀ matrix A over Q[G] (rkC is induced by C[G] → C). Known: rkG(A) ≥ rkC(A) ∀ A over Q[G] ⇒ G is locally indicable.

Andrei Jaikin-Zapirain UAM and ICMAT Homomorphisms to division rings

Parallelism between the properties of D(C[G]) and D(G)

C[G] ֒ → E(C[G]) C[G] ֒ → U(G) The division closure of C[G] D(C[G]) D(G) The elements of C[G] are

• N. I. Dubrovin
• P. Linnell

invertible if G is left-orderable (1987) (1992) The div. closure of C[G] is a Theorem A Theorem C

• div. ring if G is loc. indicable

(if is Conradian) Questions/Problems.

1 Let G be left-orderable. Are D(C[G]) and D(G) isomorphic? 2 Use analytic origin of D(G) ∼

= DC[G] in order to obtain new cases of loc. indicable groups G for which D(G) is universal.

3 Find new examples of locally indicable groups G such that

rkDC[G] = rkG(A) ≥ rkC(A) ∀ matrix A over Q[G] (rkC is induced by C[G] → C). Known: rkG(A) ≥ rkC(A) ∀ A over Q[G] ⇒ G is locally indicable.

Andrei Jaikin-Zapirain UAM and ICMAT Homomorphisms to division rings

Parallelism between the properties of D(C[G]) and D(G)

C[G] ֒ → E(C[G]) C[G] ֒ → U(G) The division closure of C[G] D(C[G]) D(G) The elements of C[G] are

• N. I. Dubrovin
• P. Linnell

invertible if G is left-orderable (1987) (1992) The div. closure of C[G] is a Theorem A Theorem C

• div. ring if G is loc. indicable

(if is Conradian) Questions/Problems.

1 Let G be left-orderable. Are D(C[G]) and D(G) isomorphic? 2 Use analytic origin of D(G) ∼

= DC[G] in order to obtain new cases of loc. indicable groups G for which D(G) is universal.

3 Find new examples of locally indicable groups G such that

rkDC[G] = rkG(A) ≥ rkC(A) ∀ matrix A over Q[G] (rkC is induced by C[G] → C). Known: rkG(A) ≥ rkC(A) ∀ A over Q[G] ⇒ G is locally indicable.

Andrei Jaikin-Zapirain UAM and ICMAT Homomorphisms to division rings

Parallelism between the properties of D(C[G]) and D(G)

C[G] ֒ → E(C[G]) C[G] ֒ → U(G) The division closure of C[G] D(C[G]) D(G) The elements of C[G] are

• N. I. Dubrovin
• P. Linnell

invertible if G is left-orderable (1987) (1992) The div. closure of C[G] is a Theorem A Theorem C

• div. ring if G is loc. indicable

(if is Conradian) Questions/Problems.

1 Let G be left-orderable. Are D(C[G]) and D(G) isomorphic? 2 Use analytic origin of D(G) ∼

= DC[G] in order to obtain new cases of loc. indicable groups G for which D(G) is universal.

3 Find new examples of locally indicable groups G such that

rkDC[G] = rkG(A) ≥ rkC(A) ∀ matrix A over Q[G] (rkC is induced by C[G] → C). Known: rkG(A) ≥ rkC(A) ∀ A over Q[G] ⇒ G is locally indicable.

Andrei Jaikin-Zapirain UAM and ICMAT Homomorphisms to division rings

Parallelism between the properties of D(C[G]) and D(G)

C[G] ֒ → E(C[G]) C[G] ֒ → U(G) The division closure of C[G] D(C[G]) D(G) The elements of C[G] are

• N. I. Dubrovin
• P. Linnell

invertible if G is left-orderable (1987) (1992) The div. closure of C[G] is a Theorem A Theorem C

• div. ring if G is loc. indicable

(if is Conradian) Questions/Problems.

1 Let G be left-orderable. Are D(C[G]) and D(G) isomorphic? 2 Use analytic origin of D(G) ∼

= DC[G] in order to obtain new cases of loc. indicable groups G for which D(G) is universal.

3 Find new examples of locally indicable groups G such that

rkDC[G] = rkG(A) ≥ rkC(A) ∀ matrix A over Q[G] (rkC is induced by C[G] → C). Known: rkG(A) ≥ rkC(A) ∀ A over Q[G] ⇒ G is locally indicable.

Andrei Jaikin-Zapirain UAM and ICMAT Homomorphisms to division rings

Thanks

THANK YOU FOR YOUR ATTENTION

Andrei Jaikin-Zapirain UAM and ICMAT Homomorphisms to division rings