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Continued from - GROUP THEORY Mathematician Disproves 80-Year-Old Algebra Conjecture - part I Olena Shmahalo
Any sweeping, universal statement about groups is almost always false, unless there’s some obvious reason why it should be true.
In many ways, the elements in a group algebra resemble the familiar polynomials from high school algebra: expressions like x2 − 4x + 5 or 3x3y5 + 2. But there’s a key difference. If you multiply two polynomials, some terms might cancel out, but the term with the highest exponent will always survive the cancellation process. For example, (x − 1)(x + 1) = x2 + x − x − 1, and while the x and −x terms cancel each other out, the x2 term survives (as does the −1), to produce x2 − 1. But in a group algebra, the relationships between group elements can lead to additional, hard-to-predict cancellations.
For example, suppose our group is the collection of symmetry transformations of the letter “A.” This group has only two elements: the transformation that leaves every point where it is (the “1” in our group), and the reflection across the central vertical axis (let’s call this reflection r). Reflecting twice restores each point to its original location, so in the language of our group multiplication, r times r equals 1. This relationship leads to all sorts of unexpected outcomes in the group algebra — for example, if you multiply r + 2 with −r/3 + 2/3, nearly everything cancels out and all that’s left is 1:
(r + 2)(−r/3 + 2/3) = −r2/3 + 2r/3 – 2r/3 + 4/3 = −r2/3 + 4/3 = 1 (since r2=1)
In other words, (r + 2) and (2/3 – r/3) are multiplicative inverses.
In 1940, an algebraist named Graham Higman made a daring conjecture in his doctoral thesis: The worst of this cancellation weirdness, he proposed, will only happen if the group that is used to construct the group algebra contains elements for which some power equals 1, as with r in the example above. In all other group algebras, he posited, while elements with just a single term, like 7a or 8b, can (and do) have multiplicative inverses, sums with multiple terms like r + 2 or 3r − 5s can never have multiplicative inverses. Since elements with multiplicative inverses are called units, Higman’s hypothesis came to be known as the unit conjecture.
Over the next few decades, Irving Kaplansky, one of the leading mathematicians of the 20th century, popularized this conjecture along with two other group algebra conjectures called the zero divisor and idempotent conjectures; the three came to be known as the Kaplansky conjectures. Collectively, the three conjectures posit that group algebras are not too radically different from the algebra we’re used to from multiplying numbers or polynomials. But although Kaplansky called attention to these conjectures, there’s no particular reason to think he believed them, Kielak said.
At the time, there was little evidence either way. If anything, there was a philosophical reason to disbelieve the conjectures: As the mathematician Mikhael Gromov is said to have observed, the menagerie of groups is so diverse that any sweeping, universal statement about groups is almost always false, unless there’s some obvious reason why it should be true.
So for Kaplansky to promote the unit conjecture was “very audacious,” Kielak said. It was “meant to provoke other people to come up with clever examples,” he said.
But mathematicians couldn’t come up with counterexamples, and not for want of trying. In the absence of a counterexample, Kielak said, “you start to think that there’s something deeper going on — that there is some underlying principle that we’ve missed.”
Collapsing Sums
Over the second half of the 20th century, a candidate for that “something deeper” seemed to emerge: algebraic K-theory, a vast edifice that uses difficult-to-compute group invariants to unite algebra with a wide range of mathematical disciplines, such as topology and number theory. Using K-theory, for instance, researchers were able to connect the unit conjecture to the question of when a topological shape can be converted into another shape using only prescribed moves.
“Powerful theory has its own beauty and elegance, but if everything is rigid, tightly controlled and well behaved, the subject can get very dry.” - Giles Gardam
Researchers were able to show that certain powerful but unproved K-theory conjectures would imply the zero divisor and idempotent conjectures, potentially offering up a deep reason why they might be true. But they couldn’t do the same for the unit conjecture, the strongest of the three. Wolfgang Lück, of the University of Bonn, tried hard to prove that the unit conjecture follows from a K-theory conjecture called the Farrell-Jones conjecture. “I was never able to make this proof,” he said. “I was wondering whether I’m stupid.”
Mathematicians were nevertheless able to prove the unit conjecture for many specific classes of groups by showing that those groups had a property akin to the notion of the highest exponent in polynomials. But researchers also knew of a handful of groups that violate this property, including a simple one called the Hantzsche-Wendt group. This group captures the symmetries of a shape physicists have considered as a possible model for the shape of the universe, and which is built by gluing together the sides of a three-dimensional crystal. Compared to many other groups, this one is “remarkably unexotic,” said Timothy Riley, of Cornell University.
The Hantzsche-Wendt group seemed like a fruitful place to search for a counterexample to the unit conjecture. But doing so was no straightforward task: The Hantzsche-Wendt group is infinite, so there are infinitely many possibilities even for short sums in the group algebra. And in 2010, a pair of mathematicians showed that if there is a counterexample in this group, it will not be found among the simplest of these sums.
Now Gardam has turned up a pair of multiplicative inverses with 21 terms each within a group algebra built from the Hantzsche-Wendt group. Finding the pair required a complex computer search, but verifying that they really are inverses is well within the realm of human computation. It’s simply a matter of multiplying them together and checking that the 441 terms in the product simplify down to the number 1. “Everything collapses down,” Kropholler said. “That’s pretty amazing.”
Lück knows now why he was never able to prove that the Farrell-Jones conjecture implies the unit conjecture: The Farrell-Jones conjecture is true for the Hantzsche-Wendt group, but the unit conjecture is false. “Now I know I wasn’t stupid,” he said.
Once Gardam releases the details of his algorithm, it will be open season for other mathematicians to explore the Hantzsche-Wendt group and potentially other groups. “The hope is that we will learn something new — a new trick which will allow us to build examples,” Kielak said.
Already, knowing that the conjecture is false has changed the mindsets of many mathematicians. “Psychologically, this is a very big difference,” Kielak said. “Probably in a year’s time, we’re going to have infinitely many” counterexamples.
Gardam’s counterexample uses one of the simplest number systems for its coefficients, a clock arithmetic with only two “hours.” So one immediate question is whether there are counterexamples to be found using other number systems such as the real or complex numbers. There’s also the question of whether some group exists that violates Kaplansky’s other two conjectures. Such a find would send shudders through the K-theory community, since it would contradict some of the subject’s central conjectures.
For Gardam, his discovery is the culmination of years spent hunting for intriguing counterexamples in algebra. He’s not motivated by a bounty hunter’s mentality, he explained in an email — rather, he chases after the frisson of delight that curious counterexamples can give.
“Powerful theory has its own beauty and elegance, but if everything is rigid, tightly controlled and well behaved, the subject can get very dry,” he wrote. “Surprising examples are a big part of what makes maths fun and keeps it weird and wonderful.”
Any sweeping, universal statement about groups is almost always false, unless there’s some obvious reason why it should be true.
In many ways, the elements in a group algebra resemble the familiar polynomials from high school algebra: expressions like x2 − 4x + 5 or 3x3y5 + 2. But there’s a key difference. If you multiply two polynomials, some terms might cancel out, but the term with the highest exponent will always survive the cancellation process. For example, (x − 1)(x + 1) = x2 + x − x − 1, and while the x and −x terms cancel each other out, the x2 term survives (as does the −1), to produce x2 − 1. But in a group algebra, the relationships between group elements can lead to additional, hard-to-predict cancellations.
For example, suppose our group is the collection of symmetry transformations of the letter “A.” This group has only two elements: the transformation that leaves every point where it is (the “1” in our group), and the reflection across the central vertical axis (let’s call this reflection r). Reflecting twice restores each point to its original location, so in the language of our group multiplication, r times r equals 1. This relationship leads to all sorts of unexpected outcomes in the group algebra — for example, if you multiply r + 2 with −r/3 + 2/3, nearly everything cancels out and all that’s left is 1:
(r + 2)(−r/3 + 2/3) = −r2/3 + 2r/3 – 2r/3 + 4/3 = −r2/3 + 4/3 = 1 (since r2=1)
In other words, (r + 2) and (2/3 – r/3) are multiplicative inverses.
In 1940, an algebraist named Graham Higman made a daring conjecture in his doctoral thesis: The worst of this cancellation weirdness, he proposed, will only happen if the group that is used to construct the group algebra contains elements for which some power equals 1, as with r in the example above. In all other group algebras, he posited, while elements with just a single term, like 7a or 8b, can (and do) have multiplicative inverses, sums with multiple terms like r + 2 or 3r − 5s can never have multiplicative inverses. Since elements with multiplicative inverses are called units, Higman’s hypothesis came to be known as the unit conjecture.
Over the next few decades, Irving Kaplansky, one of the leading mathematicians of the 20th century, popularized this conjecture along with two other group algebra conjectures called the zero divisor and idempotent conjectures; the three came to be known as the Kaplansky conjectures. Collectively, the three conjectures posit that group algebras are not too radically different from the algebra we’re used to from multiplying numbers or polynomials. But although Kaplansky called attention to these conjectures, there’s no particular reason to think he believed them, Kielak said.
At the time, there was little evidence either way. If anything, there was a philosophical reason to disbelieve the conjectures: As the mathematician Mikhael Gromov is said to have observed, the menagerie of groups is so diverse that any sweeping, universal statement about groups is almost always false, unless there’s some obvious reason why it should be true.
So for Kaplansky to promote the unit conjecture was “very audacious,” Kielak said. It was “meant to provoke other people to come up with clever examples,” he said.
But mathematicians couldn’t come up with counterexamples, and not for want of trying. In the absence of a counterexample, Kielak said, “you start to think that there’s something deeper going on — that there is some underlying principle that we’ve missed.”
Collapsing Sums
Over the second half of the 20th century, a candidate for that “something deeper” seemed to emerge: algebraic K-theory, a vast edifice that uses difficult-to-compute group invariants to unite algebra with a wide range of mathematical disciplines, such as topology and number theory. Using K-theory, for instance, researchers were able to connect the unit conjecture to the question of when a topological shape can be converted into another shape using only prescribed moves.
“Powerful theory has its own beauty and elegance, but if everything is rigid, tightly controlled and well behaved, the subject can get very dry.” - Giles Gardam
Researchers were able to show that certain powerful but unproved K-theory conjectures would imply the zero divisor and idempotent conjectures, potentially offering up a deep reason why they might be true. But they couldn’t do the same for the unit conjecture, the strongest of the three. Wolfgang Lück, of the University of Bonn, tried hard to prove that the unit conjecture follows from a K-theory conjecture called the Farrell-Jones conjecture. “I was never able to make this proof,” he said. “I was wondering whether I’m stupid.”
Mathematicians were nevertheless able to prove the unit conjecture for many specific classes of groups by showing that those groups had a property akin to the notion of the highest exponent in polynomials. But researchers also knew of a handful of groups that violate this property, including a simple one called the Hantzsche-Wendt group. This group captures the symmetries of a shape physicists have considered as a possible model for the shape of the universe, and which is built by gluing together the sides of a three-dimensional crystal. Compared to many other groups, this one is “remarkably unexotic,” said Timothy Riley, of Cornell University.
The Hantzsche-Wendt group seemed like a fruitful place to search for a counterexample to the unit conjecture. But doing so was no straightforward task: The Hantzsche-Wendt group is infinite, so there are infinitely many possibilities even for short sums in the group algebra. And in 2010, a pair of mathematicians showed that if there is a counterexample in this group, it will not be found among the simplest of these sums.
Now Gardam has turned up a pair of multiplicative inverses with 21 terms each within a group algebra built from the Hantzsche-Wendt group. Finding the pair required a complex computer search, but verifying that they really are inverses is well within the realm of human computation. It’s simply a matter of multiplying them together and checking that the 441 terms in the product simplify down to the number 1. “Everything collapses down,” Kropholler said. “That’s pretty amazing.”
Lück knows now why he was never able to prove that the Farrell-Jones conjecture implies the unit conjecture: The Farrell-Jones conjecture is true for the Hantzsche-Wendt group, but the unit conjecture is false. “Now I know I wasn’t stupid,” he said.
Once Gardam releases the details of his algorithm, it will be open season for other mathematicians to explore the Hantzsche-Wendt group and potentially other groups. “The hope is that we will learn something new — a new trick which will allow us to build examples,” Kielak said.
Already, knowing that the conjecture is false has changed the mindsets of many mathematicians. “Psychologically, this is a very big difference,” Kielak said. “Probably in a year’s time, we’re going to have infinitely many” counterexamples.
Gardam’s counterexample uses one of the simplest number systems for its coefficients, a clock arithmetic with only two “hours.” So one immediate question is whether there are counterexamples to be found using other number systems such as the real or complex numbers. There’s also the question of whether some group exists that violates Kaplansky’s other two conjectures. Such a find would send shudders through the K-theory community, since it would contradict some of the subject’s central conjectures.
For Gardam, his discovery is the culmination of years spent hunting for intriguing counterexamples in algebra. He’s not motivated by a bounty hunter’s mentality, he explained in an email — rather, he chases after the frisson of delight that curious counterexamples can give.
“Powerful theory has its own beauty and elegance, but if everything is rigid, tightly controlled and well behaved, the subject can get very dry,” he wrote. “Surprising examples are a big part of what makes maths fun and keeps it weird and wonderful.”