Gemara

59a (Mishnah): A Cheresh gestures or reacts to gestures (to make transactions);

Ben Beseirah says, he is Kofetz or responds to Kefitzah for Metaltelim.

(Rav Nachman) Version #1: They argue only about Metaltelim, but all agree that gesturing suffices for Gitin.

Version #2: Just like they argue about Metaltelim, they also argue about Gitin.

67b (Mishnah): If a man became unable to speak and others asked him if we should write a Get for his wife and he nodded his head, we test him three times. If he answers properly no and yes, we write and give the Get.

71a (R. Zeira): The Mishnah discusses a mute. He is not like a (total) Cheresh:

(Beraisa): One who can speak but not hear is a (partial) Cheresh. One who can hear but not speak is a mute. Both of them are like Pikchim (sound-minded people) in every respect.

(Beraisa): Just like we test one who became mute (to see if he is alert enough) to give a Get, we also test regarding sales and gifts, testimony, and inheritances.

This means that we test for what he sells and gives.

(Beraisa): Regarding Metaltelim we rely on gestures and Kefitzah (lip movements) of a Cheresh and what he writes, but not for Gitin.

Rishonim

The Rif and Rosh (Gitin 27b and 5:18) bring the Mishnah (Gitin 59a) and both versions of Rav Nachman.

Rosh (ibid.): The Rif brings Rav Nachman's teaching, i.e. for he rules like Ben Beseira. He is lenient like Version #2. I disagree. Throughout Shas the Amora'im explain Tana'im argue in order to understand the Mishnah, even if the Halachah does not follow the Tana they explain. One cannot be lenient without a proof.

Rebuttal (Tif'eres Shmuel 1): The Rif does not rule like Rav Nachman. He brings both versions to teach that Gitin are like Metaltelim; gesturing suffices. This is unlike a Beraisa (71a) that does not allow relying on gestures for Gitin.

Defense (Korban Nesan'el 20): Surely, just like he married through gesturing, he may divorce by gesturing! That Beraisa applies to Yibum!

Korban Nesan'el (30): The Rosh himself often rules like an opinion because the Gemara explained it. This refers to a Beraisa or Amora'im. The Gemara explains opinions in the Mishnah even if the Halachah does not follow them.

Beis Yosef (CM 235 DH ha'Cheresh and EH 121 DH u'Mah she'Chosav Rabeinu): The Rosh says that the Rif holds that Kefitzah is weaker than gesturing, and the Halachah follows Ben Beseira. He must explain that the Mishnah (Yevamos 112b) discusses marriage and divorce through gesturing, even though Kefitzah suffices, for all agree to this. The Rambam mentions only gesturing. He learns from that Stam Mishnah. Perhaps the Rif agrees. He cited Rav Nachman only to show that we require gesturing also for Gitin. We are not lenient to allow Kefitzah to prevent Igun (inability to remarry).

Rif and Rosh (Gitin 34a and 6:3): We rely on a Cheresh's gestures, Kefitzah and what he writes for Metaltelim, but not for Gitin. This refers to a Pike'ach who became deaf after marriage. If a Cheresh married, we check him. If he wants, we divorce his wife. If a Cheresh married a Pikachas or vice-versa, he can choose to divorce her by gesturing, just like he married by gesturing (Yevamos 112b). If a Pike'ach married a Pikachas and she went deaf, he may divorce her, but not if she went insane. If he went deaf or insane, he may never divorce her. A Mishnah teaches that a Cheresh gestures or reacts to gestures. Ben Beseirah says, he is Kofetz or responds to Kefitzah for Metaltelim. Just like they argue about Metaltelim, they also argue about Gitin.

Ran (DH Romez): Gesturing is with the hand and head. Kefitzah is with the lips; it is less clear than gesturing.

Aruch (Erech Remez): Remizah (gesturing) is with the eye or hand. It is the Targum of "Koretz Ayin" and "Sholach Etzba".

Rambam (Hilchos Mechirah 29:2): One who cannot hear or speak, or if he can speak but cannot hear at all, can buy and sell Metaltelim through gestures, but not land. Even for Metaltelim, we must test him well before validating his actions.

Question (Ra'avad): I do not know why (this is not for land).

Answer (Migdal Oz): Chachamim enacted that Pa'utos (mature children) can buy and sell Metaltelim, for their needs of living. The Gemara said this only about Pa'utos, but the same applies to a Cheresh. We always equate a child, lunatic and Cheresh. The Ramban says that Chachamim enacted for common cases. Almost every heir inherits Metaltelim, but not all inherit land. The Gemara makes no allusion to this.

Rambam (3): If one can hear but cannot speak, or if a person became mute, his purchase, sale and gift of land or Metaltelim is valid. We must test him like we test for Gitin, or he himself must write (what he wants to do).

Question (Magid Mishnah 2): Actions of a Cheresh, i.e. one who cannot hear or speak are valid through gestures (59a). He is no worse than one who became mute, whom we test to see if he can authorize a Get. However, the Gemara says that one who can speak but not hear or vice-versa is like a Pike'ach in every way. It is not clear why the Rambam distinguishes them.

Answer (Kesef Mishneh 2): The Rambam learns from 59a, which discusses whether or not Metaltelim excludes Gitin. It does not discuss land. This implies that land is like Gitin. Since we rely on nodding or writing for Gitin, the same applies to land. The Gemara cannot mean that one who can speak or hear, but not both, is exactly like a Pike'ach. We need not test a Pike'ach! Rather, he is not a total Cheresh. Alternatively, one who can speak but not hear is like a total Pike'ach, i.e. he can hear a little, but not like Pikchim. This is why the Rambam (Halachah 2) considers one who can speak but cannot hear at all like a total Cheresh.

Rebuttal (Nesivos ha'Mishpat 20): A Cheresh is one who does not hear or speak (Chagigah 2b). This implies that one who can speak is a Pike'ach! If he hears a little he is a Pike'ach even if he cannot talk! Why didn't the Rambam teach that if one can speak but cannot hear, he cannot Mekadesh or divorce? Rather, the Rambam holds like Rashi; Cheresh refers (only) to a deaf-mute only when it is taught together with a lunatic and child. Only one who can hear can sell land, for this requires great sharpness. Many do not allow even a Pike'ach to sell land until he is 20 years old!

Poskim

Shulchan Aruch (CM 235:17): If a Cheresh cannot hear or speak, or if he can speak but cannot hear at all, he can buy and sell Metaltelim through gestures, but not land. Even for Metaltelim we must test him well before validating his actions.

Beis Yosef (DH v'Echad): If one became mute, we must test him before writing a Get for him (67b). Just like we test one who became mute to give a Get, also for sales and gifts (71a). All the more so, a Cheresh who cannot hear must be tested for these!

Beis Yosef (DH u'Mah she'Chosav v'Yir'eh and DH u'Mah she'Chosav v'Ein): The Tur says that one who can speak but not hear is like a Pike'ach. We can tell from what he says whether or not his mind is sound.

Bach (CM 235 DH u'Mah she'Chosav v'Ein): The Rambam understands that the Mishnah (59a) calls a Cheresh anyone who cannot hear, even if he can speak, like the Beraisa (71a). The Tur holds that the Beraisa (71a, we rely on gestures... for Metaltelim, but not for Gitin) discusses even a Cheresh who can speak. It omits one who became mute, for he can sell even land.

Shulchan Aruch (18): If one is mute (he can hear but cannot speak) or he became mute, his purchase, sale and gift of land or Metaltelim is valid. We must test him like we test for Gitin, or he himself must write (what he wants to do).

SMA (49): Mute refers to one who never talked, from infancy. One who became mute means that he once talked.