Timeline for About Lie group $G$ has this escape property？
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28 events
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| Aug 21, 2021 at 14:56 | comment | added | Carl-Fredrik Nyberg Brodda | @ya_yang He responded 10 minutes after you posted the comment. If you did not think anyone read those comments, why did you post it? | |
| Aug 20, 2021 at 20:52 | comment | added | Carl-Fredrik Nyberg Brodda | Is this not the exact question you asked Terence Tao on his blog, in this comment? Was the answer there not satisfactory (it is the same as Dmitry Vaintrob's answer here)? | |
| Aug 20, 2021 at 20:32 | history | edited | free | CC BY-SA 4.0 |
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| Aug 20, 2021 at 14:09 | history | edited | free | CC BY-SA 4.0 |
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| Aug 20, 2021 at 13:56 | history | edited | free | CC BY-SA 4.0 |
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| Aug 20, 2021 at 8:06 | comment | added | Andrea Marino | @Mark Sapir: yes, I was wrong. I only understood now what you meant by "a, b can be too close": in the range in which we can control the evolution via exponential map, $a^m, b^m$ could still be too close, and then we lose control. Anyway, we are plenty of counterexamples at the moment!! :) | |
| Aug 20, 2021 at 7:17 | answer | added | YCor | timeline score: 3 | |
| Aug 20, 2021 at 6:36 | comment | added | free | @ Dmitry Vaintrob:Thanks for your comments | |
| Aug 20, 2021 at 6:32 | answer | added | Dmitry Vaintrob | timeline score: 4 | |
| Aug 20, 2021 at 6:18 | history | edited | YCor | CC BY-SA 4.0 |
fixed English, changed tag
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| Aug 20, 2021 at 6:13 | comment | added | free | I still don't solve this question Through the above way . Because Baker-Hausdorff formulas is true for small neighbor hood. As m tend to infinite, the definition of logarithm map can't be given. | |
| Aug 20, 2021 at 6:11 | comment | added | Dmitry Vaintrob | If $G$ is any compact noncommutative connected group, this is not true. Indeed, for a compact group you can define an equivariant distance function on $G$ such that $d(a,b) = d(1, ab^{-1}).$ The triangle inequality then implies $$d(X, gXg^{-1}) \le 2 d(1, g).$$ for any element $X\in G.$ But this means that if $a$ is any element in $U$ and $g$ is small enough that the ball around $I$ of radius $2d(1,g)$ is contained in $U$, then $ga^ng^{-1}\in U.$ Thus if $a, b$ are nearby conjugate elements, the conjectured property fails. | |
| Aug 20, 2021 at 5:42 | history | edited | Asaf Karagila♦ | CC BY-SA 4.0 |
appended answer 402097 as supplemental
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| Aug 20, 2021 at 1:42 | comment | added | free | @AndreaMarino: I feel like your way is true. I will try to do it | |
| Aug 20, 2021 at 1:10 | comment | added | free | @Mark Sapir :The question was not raised. It's a strange question if it true or wrong. | |
| Aug 20, 2021 at 0:12 | comment | added | markvs | @AndreaMarino: So what's the proof? As far as I see, the claim is not proved and may even be wrong. | |
| Aug 19, 2021 at 22:44 | comment | added | Andrea Marino | If we place ourselves in the neighborhood $U$ Buzz refers to and set $a = e^{\alpha}, b = e^{\beta}$, we can express $a^m (b^{-1})^m = e^{m \alpha} e^{-m \beta} = e^{F(m\alpha, -m \beta)}$, where the big $F$ is given by the Baker-Campbell-Hausdorff formula. Let $U$ correspond to the ball of the tangent space of radius $r$. I think we now have to show that as $m$ goes to $\infty$, the norm of $F(m \alpha, m \beta)$ goes to $\infty$ for all $\alpha, \beta$ of norm $\le r$.. In this way $a^m (b^{-1})^m$ can't be in $U$, otherwise its logarithm would have radius $\le r$. | |
| Aug 19, 2021 at 21:33 | comment | added | markvs | @Buzz: Your suggestion may not work because $a^m$ may be too close to $b^m$. | |
| S Aug 19, 2021 at 21:09 | history | edited | Johannes Hahn | CC BY-SA 4.0 |
Fixed grammar
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| S Aug 19, 2021 at 21:09 | history | suggested | markvs | CC BY-SA 4.0 |
Fixed misprints
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| Aug 19, 2021 at 20:31 | comment | added | free | Thanks your suggestion. | |
| Aug 19, 2021 at 20:23 | comment | added | Buzz | The exponential mapping always maps provides a diffeomorphism from a sufficiently small ball around the origin in the tangent space (the Lie algebra $\mathfrak{g}$) to a neighborhood of $e$ in $G$. This gives the logarithmic coordinates for $G$ around the identity, which it seems like should be enough to prove this. | |
| Aug 19, 2021 at 19:51 | review | Suggested edits | |||
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| Aug 19, 2021 at 19:50 | history | edited | Michael Albanese | CC BY-SA 4.0 |
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| Aug 19, 2021 at 19:41 | review | First posts | |||
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| Aug 19, 2021 at 19:34 | history | asked | free | CC BY-SA 4.0 |